title: George Washington's Farewell Address

text: The thought of the United States without George Washington as its president caused concern among many Americans. Thomas Jefferson disagreed with many of the president's policies and later led the Democratic-Republicans in opposition to many Federalist policies, but he joined his political rival Alexander Hamilton—the leader of the Federalists—in convincing the president to delay his retirement and serve a second term. The two men feared that the nation would be torn apart without his leadership. Washington most likely referred to this when he told the American people that he had wanted to retire before the last election, but he was convinced by people who were, in his words, "entitled to my confidence" that it was his duty to serve a second term. All of the ideas presented in Washington's Farewell Address came from Washington; however, Alexander Hamilton wrote most of it.


title: George Washington's Farewell Address

text: Washington sought to convince the American people that his service was no longer necessary by telling them, as he had in his first inaugural address, that he truly believed that he was never qualified to be president. If he accomplished anything during his presidency, he said, it was as a result of their support and efforts to help the country survive and prosper. Despite his confidence that the country would survive without his leadership, Washington used the majority of the letter to offer advice as a "parting friend" on what he believed were the greatest threats to the nation.


title: George Washington's Farewell Address

text: The Address expresses Washington's understanding of republicanism by affirming popular government and warning about threats to "Republican liberty". He begins his warnings to the American people by stressing that their independence, peace at home and abroad, safety, prosperity, and liberty are all dependent upon unity among the states. He warns them that the union of states created by the Constitution will come under the most frequent and focused attacks by foreign and domestic enemies of the country. He warns the American people to be suspicious of anyone who seeks to abandon the Union, secede a portion of the country from the rest, or weaken the bonds that hold together the constitutional union. To promote the strength of the Union, he urges the people to place their identity as Americans above their identities as members of a state, city, or region, and to focus their efforts and affection on the country above all other local interests. He reminds the people that they do not have more than slight differences in religion, manners, habits, and political principles and that their triumph and possession of independence and liberty are the results of working together.


title: George Washington's Farewell Address

text: Washington continues to express his support of the Union by giving some examples of how he believes that the country, its regions, and its people are already benefiting from the unity which they currently share. He then looks to the future in his belief that the combined effort and resources of its people will protect the country from foreign attack and allow them to avoid wars between neighboring nations that often happen due to rivalries and competing relations with foreign nations. He argues that the security provided by the Union will also allow the United States to avoid the creation of an overgrown military which he sees as a great threat to liberty, especially the republican liberty that the United States has created.


title: George Washington's Farewell Address

text: Washington goes on to warn the American people to question the ulterior motives of any person or group who argues that the land within the borders of the United States is too large to be ruled as a republic, an argument made by many during the debate on the proposed purchase of the Louisiana Territory, calling on the people to give the experiment of a large republic a chance to work before deciding that it cannot be done. He then offers strong warnings on the dangers of sectionalism, arguing that the true motives of a sectionalist are to create distrust or rivalries between regions and people to gain power and take control of the government. Washington points to the Jay Treaty and Pinckney's Treaty which established the borders of the United States western territories between Spanish Mexico and British Canada and secured the rights of western farmers to ship goods along the Mississippi River to New Orleans. He holds up these treaties as proof that the eastern states along the Atlantic Coast and the federal government are looking out for the welfare of all the American people and can win fair treatment from foreign countries as a united nation.


title: George Washington's Farewell Address

text: Washington goes on to state his support for the new constitutional government, calling it an improvement upon the nation's original attempt in the Articles of Confederation. He reminds the people that it is the right of the people to alter the government to meet their needs, but it should only be done through constitutional amendments. He reinforces this belief by arguing that violent takeovers of the government should be avoided at all costs and that it is the duty of every member of the republic to follow the constitution and to submit to the laws of the government until it is constitutionally amended by the majority of the American people.


title: George Washington's Farewell Address

text: Washington warns the people that political factions may seek to obstruct the execution of the laws created by the government or to prevent the branches of government from exercising the powers provided to them by the constitution. Such factions may claim to be trying to answer popular demands or solve pressing problems, but their true intentions are to take the power from the people and place it in the hands of unjust men.


title: George Washington's Farewell Address

text: Washington calls the American people to only change the Constitution through amendments, but he then warns them that groups seeking to overthrow the government may strive to pass constitutional amendments to weaken the government to a point where it is unable to defend itself from political factions, enforce its laws, and protect the people's rights and property. As a result, he urges them to give the government time to realize its full potential, and only amend the constitution after thorough time and thought have proven that it is truly necessary instead of simply making changes based upon opinions and hypotheses of the moment...


title: George Washington's Farewell Address

text: Washington continues to advance his idea of the dangers of sectionalism and expands his warning to include the dangers of political parties to the country as a whole. These warnings are given in the context of the recent rise of two opposing parties within the government—the Democratic-Republican Party led by Jefferson, and Hamilton's Federalist Party. Washington had striven to remain neutral during a conflict between Britain and France brought about by the French Revolution, while the Democratic-Republicans had made efforts to align with France, and the Federalists had made efforts to ally with Great Britain.


title: George Washington's Farewell Address

text: Washington recognizes that it is natural for people to organize and operate within groups such as political parties, but he also argues that every government has recognized political parties as an enemy and has sought to repress them because of their tendency to seek more power than other groups and to take revenge on political opponents. He feels that disagreements between political parties weakened the government.


title: George Washington's Farewell Address

text: Moreover, he makes the case that "the alternate domination" of one party over another and coinciding efforts to exact revenge upon their opponents have led to horrible atrocities, and "is itself a frightful despotism. But this leads at length to a more formal and permanent despotism." From Washington's perspective and judgment, political parties eventually and "gradually incline the minds of men to seek security… in the absolute power of an individual", leading to despotism. He acknowledges the fact that parties are sometimes beneficial in promoting liberty in monarchies, but he argues that political parties must be restrained in a popularly elected government because of their tendency to distract the government from their duties, create unfounded jealousies among groups and regions, raise false alarms among the people, promote riots and insurrection, and provide foreign nations and interests access to the government where they can impose their will upon the country.


title: George Washington's Farewell Address

text: Washington continues his defense of the Constitution by stating that the system of checks and balances and separation of powers within it are important means of preventing a single person or group from seizing control of the country. He advises the American people that, if they believe that it is necessary to modify the powers granted to the government through the Constitution, it should be done through constitutional amendments instead of through force.


title: George Washington's Farewell Address

text: One of the most referenced parts of Washington's letter is his strong support of the importance of religion and morality in promoting private and public happiness and in promoting the political prosperity of the nation. He argues that religious principles promote the protection of property, reputation, life, and honor that are the foundations of justice. He cautions against the belief that the nation's morality can be maintained without religion:


title: George Washington's Farewell Address

text: Washington refers to religious principle as the foundation of a civilized society. He also argues that the American government needs to ensure "the general diffusion of knowledge" throughout the United States; the government has been created to enforce the opinion of the people, so the opinion of the people should be informed and knowledgeable.


title: George Washington's Farewell Address

text: Washington provides strong support for the balanced federal budget, arguing that the nation's credit is an important source of strength and security. He urges the American people to preserve the national credit by avoiding war, avoiding unnecessary borrowing, and paying off any national debt accumulated in times of war as quickly as possible in times of peace so that future generations do not have to take on the financial burdens. Despite his warnings to avoid taking on debt, Washington does state his belief that sometimes it is necessary to spend money to prevent dangers or wars that will cost more if not properly prepared for. At these times, he argues, it is necessary for the people to cooperate by paying taxes to cover these precautionary expenses. He emphasizes how important it is for the government to be careful in choosing the items that will be taxed, but also ball the American people that, no matter how hard the government tries, there will never be a tax which is not inconvenient and unpleasant to those who must pay it.


title: George Washington's Farewell Address

text: Washington dedicates a large part of his farewell address to discussing foreign relations and the dangers of permanent alliances between the United States and foreign nations, which he views as foreign entanglements. He advocates a policy of good faith and justice towards all nations, again making reference to proper behavior based upon religious doctrine and morality. He urges the American people to avoid long-term friendly relations or rivalries with any nation, arguing that attachments with or animosity toward other nations will only cloud the government's judgment in its foreign policy. He argues that longstanding poor relations will only lead to unnecessary wars due to a tendency to blow minor offenses out of proportion when committed by nations viewed as enemies of the United States. He continues this argument by claiming that alliances are likely to draw the United States into wars that have no justification and no benefit to the country beyond simply defending the favored nation. Alliances, he warns, often lead to poor relations with nations who feel that they are not being treated as well as America's allies, and threaten to influence the American government into making decisions based upon the will of their allies instead of the will of the American people.


title: George Washington's Farewell Address

text: Washington makes an extended reference to the dangers of foreign nations who will seek to influence the American people and government; nations who may be considered friendly as well as nations considered enemies will equally try to influence the government to do their will. "Real patriots", he warns, who "resist the intrigues" of foreign nations may find themselves "suspected and odious" in the eyes of others, yet he urges the people to stand firm against such influences all the same. He portrays those who attempt to further such foreign interests as becoming the "tools and dupes" of those nations, stealing the applause and praise of their country away from the "real patriots" while actually working to "surrender" American interests to foreign nations.


title: George Washington's Farewell Address

text: Washington goes on to urge the American people to take advantage of their isolated position in the world, and to avoid attachments and entanglements in foreign affairs, especially those of Europe, which he argues have little or nothing to do with the interests of America. He argues that it makes no sense for the American people to become embroiled in European affairs when their isolated position and unity allow them to remain neutral and focus on their own affairs. He argues that the country should avoid permanent alliances with all foreign nations, although temporary alliances during times of extreme danger may be necessary. He states that current treaties should be honored but not extended.


title: George Washington's Farewell Address

text: Washington wraps up his foreign policy stance by advocating free trade with all nations, arguing that trade links should be established naturally and the role of the government should be limited to ensuring stable trade, defending the rights of American merchants and any provisions necessary to ensure the conventional rules of trade.


title: George Washington's Farewell Address

text: Washington uses this portion of the address to explain that he does not expect his advice to make any great impression upon the people or to change the course of American politics, but he does hope that the people will remember his devoted service to his country.


title: George Washington's Farewell Address

text: Washington then explains his reasoning behind the Proclamation of Neutrality which he made during the French Revolutionary Wars, despite the standing Treaty of Alliance with France. He explains that the United States had a right to remain neutral in the conflict and that the correctness of that decision "has been virtually admitted by all" nations since. Justice and humanity required him to remain neutral during the conflict, he argues, and the neutrality was also necessary to allow the new government a chance to mature and gain enough strength to control its own affairs.


title: George Washington's Farewell Address

text: Washington closes his letter to the American people by asking them to forgive any failures which may have occurred during his service to the country, assuring them that they were due to his own weaknesses and by no means intentional. The sentences express his excitement about joining his fellow Americans as a private citizen in the free government which they have created together during his 45 years of public service.


title: George Washington's Farewell Address

text: Washington's Farewell Address is considered to be one of the most important documents in American history and the foundation of the Federalist Party's political doctrine.


title: George Washington's Farewell Address

text: Washington later accepted a commission from President John Adams, despite his stated desire to retire from public service, as the Senior Officer of a Provisional Army formed to defend the nation against a possible invasion by French forces during the Quasi-War. Washington held true to his statements in his farewell address, despite spending months organizing the Officer Corps of the Provisional Army, and declined suggestions that he return to public office in the presidential election of 1800.


title: George Washington's Farewell Address

text: Washington's statements on the importance of religion and morality in American politics and his warnings on the dangers of foreign alliances influenced political debates into the twentieth century, and have received special consideration as advice from an American hero.


title: George Washington's Farewell Address

text: Washington's hope that the United States would end permanent alliances with foreign nations was realized in 1800 with the Convention of 1800, the Treaty of Mortefontaine which officially ended the 1778 Treaty of Alliance, in exchange for ending the Quasi-War and establishing most favored nation trade relations with Napoleonic France. In 1823, Washington's foreign policy goals were further realized in the Monroe Doctrine, which promised non-interference in European affairs so long as the nations of Europe did not seek to colonize or interfere with the newly independent Latin American nations of Central and South America. The United States did not enter into any permanent military alliances with foreign nations until the 1949 North Atlantic Treaty which formed NATO.


title: George Washington's Farewell Address

text: Philadelphia residents signed a petition in January 1862 during the American Civil War requesting that Congress commemorate the 130th anniversary of Washington's birth by reading his Farewell Address "in one or the other of the Houses of Congress." It was first read in the House of Representatives in February 1862, and reading Washington's address became a tradition in both houses by 1899. The House of Representatives abandoned the practice in 1984, but the Senate continues the tradition. Washington's Birthday is observed by selecting a member of the Senate to read the address aloud on the Senate floor, alternating between political parties each year since 1896. Additionally, readers make an entry into a black, leather-bound journal maintained by the Secretary of the Senate upon finishing.


title: George Washington's Farewell Address

text: During Strom Thurmond's filibuster of the Civil Rights Act of 1957, George Washington's Farewell Address was read by Strom Thurmond.


title: George Washington's Farewell Address

text: According to political journalist John Avlon, the Farewell Address was "once celebrated as a civic Scripture, more widely reprinted than the Declaration of Independence" but adds that it "is now almost forgotten." He suggested that it had long been "eclipsed in the national memory" until the Broadway musical "Hamilton" brought it back to popular awareness in the song "One Last Time", where lines are sung by Washington and Hamilton from the end of the Address.


title: María Félix

text: María de los Ángeles Félix Güereña (; 8 April 1914 – 8 April 2002) was a Mexican actress and singer. Along with Pedro Armendáriz and Dolores del Río, she was one of the most successful figures of Latin American cinema in the 1940s and 1950s. Considered one of the most beautiful actresses of the Golden Age of Mexican cinema, her taste for the finesse and strong personality garnered her the title of diva early in her career. She was known as "La Doña", a name derived from her character in "Doña Bárbara" (1943), and "María Bonita", thanks to the anthem composed exclusively for her as a wedding gift by Félix's second husband, Agustín Lara. Her acting career consists of 47 films made in Mexico, Spain, France, Italy and Argentina.


title: María Félix

text: María de los Ángeles Félix Güereña was born in Álamos, Sonora, Mexico, on 8 April 1914. Her birth was registered on May 4, which was later misreported as her date of birth.


title: María Félix

text: She was the daughter of Bernardo Félix Flores, a military officer and politician who was of distant Yaqui descent, and Josefina Güereña Rosas, who grew up in the U.S. state of California; both were of Basque Mexican ancestry. She had fifteen siblings: Josefina, María de la Paz, Pablo, Bernardo, Miguel, María de las Mercedes, Fernando, Victoria Eugenia, Ricardo, Benjamín and Ana María del Sacramento.


title: María Félix

text: She spent her childhood in Álamos. During her childhood, she had a close relationship with her brother Pablo. Her mother separated the two siblings, thinking they might be involved in an incestuous relationship. For that reason, Pablo was sent to the Colegio Militar, in Mexico City.


title: María Félix

text: Later, the Félix family moved to Guadalajara. When María was 17, her beauty soon began to attract attention. She was crowned Beauty Queen at the University of Guadalajara. It was at this time that she met Enrique Álvarez Alatorre, a salesman for the cosmetics firm Max Factor. After a brief romance, the couple married in 1931. In 1935, Félix gave birth to her only child, Enrique, nicknamed "Quique". Her marriage with Álvarez was unsuccessful and the couple divorced in 1937. After her divorce, Félix returned to Guadalajara with her family, where she was the subject of gossip and rumors due to her status as a divorcée. Because of this situation, Félix decided to move to Mexico City with her son.


title: María Félix

text: In Mexico City, she worked as a receptionist in a plastic surgeon's office, and lived in a guest house. One day, the father of her son visited the child, and deliberately refused to give the boy back to his mother. Álvarez took the child to Guadalajara. Félix's son was recovered with the help of Agustín Lara, her second husband. They planned an elaborate recovery that tricked the grandmother and took the child.


title: María Félix

text: One afternoon after work when walking down the street in Mexico City, director and filmmaker Fernando Palacios approached her asking if she wanted to make movies. Her response was:


title: María Félix

text: Palacios finally persuaded her to break into the movies. Becoming her "Pygmalion", he began to train her and present her in film circles. She made her first appearance in the "White and Black Ballroom" of the Mexico City Country Club where some of the great Mexican movie stars of the era (Esther Fernández, Lupe Vélez, Andrea Palma) gathered. Eventually she was taken to Hollywood, to the Metro-Goldwyn-Mayer Studios, where she met Cecil B. DeMille, who offered to launch her film career in Hollywood, but Félix was not interested. She preferred to begin her career in her own country. Finally, thanks to Palacios, she was offered the female lead role in a film by Grovas Productions: "El Peñón de las Ánimas", directed by Miguel Zacarías.


title: María Félix

text: In the film "El Peñón de las Ánimas", Félix starred opposite the popular Mexican actor and singer Jorge Negrete. María Félix and Jorge Negrete got off to a bad relationship during the filming because he had asked that his girlfriend, the actress Gloria Marín, be given the lead role. For this reason the filming of the movie was difficult and led to a direct confrontation between Félix and Negrete. That confrontation helped to cement the reputation of Félix as a tough and arrogant woman.


title: María Félix

text: In her second film, "María Eugenia", Félix would be projected in a role out of her temperamental film personality. However, the film is remembered as the only movie where Félix appeared in swimwear. The same happened with the film "La china poblana" (1943), where Félix claimed to have paid a debt of gratitude to her discoverer Fernando Palacios, who directed the film. Nobody has heard from this movie (the second color film of the Mexican cinema), so it is considered lost. She called both films "her beginner sins".


title: María Félix

text: Félix was known as "La Doña" for her role in the movie "Doña Bárbara" (1943), based on the novel by the Venezuelan writer Rómulo Gallegos. For the film, another actress (Isabela Corona) was already hired, but when Gallegos first saw Félix, he was charmed by her and said: "Here is my Doña Bárbara!" This movie was filmed in Venezuela with Mexican and Venezuelan actors. "Doña Bárbara" was to be the start of her major collaborations with the Mexican film director Fernando de Fuentes. Félix and de Fuentes filmed together another two films: "La Mujer sin Alma" (1944) and "La Devoradora" (1946). As María Félix herself said, ""With these films, I became the number one enemy of the Mexican family morals.""


title: María Félix

text: Without her stereotyped role of a "femme fatale", María Félix filmed two sophisticated films: "El monje blanco" (1945), directed by Julio Bracho, and "Vertigo" (1946), directed by Antonio Momplet. Both characters required a great dramatic intensity.


title: María Félix

text: Under the direction of Emilio Fernández, María Félix made three successful films: "Enamorada" (1946), "Río Escondido" (1947), and "Maclovia" (1948). The relationship between Félix and Fernández was cordial and smooth, despite the strong and famous temperament of the film director. In "Enamorada", María Félix found her perfect film partner, the actor Pedro Armendáriz. The films of María Félix with Fernández and his team (writer Mauricio Magdaleno, photographer Gabriel Figueroa, and Armendáriz) had strong presence in several international film festivals. In turn, they gave María Félix her first Ariel Award for "Enamorada" and "Río Escondido".


title: María Félix

text: Between the films with Fernández, María Félix also worked with Roberto Gavaldón, another director who showcased some of her best performances. Their first collaboration was in "La diosa arrodillada" (1947) with Arturo de Córdova. Thanks to these films, María Félix's fame crossed the Atlantic.


title: María Félix

text: In 1948 she was contracted by the Spanish film producer Cesáreo González. In this manner María Félix began her film adventure in Europe. She made only one more film in Mexico ("Doña Diabla", 1949) until her return in 1952.


title: María Félix

text: Félix debuted in the European cinema in Spain, with the film "Mare Nostrum" (1948), directed by Rafael Gil. With Gil as director, she made another two films in 1950: "Una mujer cualquiera" and "La noche del sábado". In 1951, she filmed the French-Spanish production "La Couronne Noire" directed by Luis Saslavsky based on a story by Jean Cocteau. She debuted in Italy with the film "Incantesimo Tragico" (1951). In the same year, she filmed "Messalina" (1951), directed by Carmine Gallone (at the time, the most expensive film of Italian cinema). During the filming of this movie Félix suffered an emotional shock: her father died of a heart attack in Navojoa.


title: María Félix

text: In 1952, Félix traveled to Argentina, where she filmed "La pasión desnuda" of the Argentine filmmaker Luis César Amadori. In the same year Félix returned to Mexico. She concluded her working relationship with Cesáreo González with the film "Camelia" filmed in her native country. Her stay in Mexico was mainly based on her marriage to the actor and singer Jorge Negrete, her former film rival, with whom she filmed "El rapto", directed by Emilio Fernández, Negrete's last film.


title: María Félix

text: After Negrete's death on 5 December 1953, Félix returned to Europe. In France she made the films "La Belle Otero" (1954), and "Les Héros sont Fatigués" (1955), alongside Yves Montand. However, the most important film of Félix in this period was "French Cancan" (1954) directed by Jean Renoir with the legendary French actor Jean Gabin. Her last film shot entirely in Europe was the Spanish film "Faustina".


title: María Félix

text: Félix returned to Mexico in 1955. This period of her career was characterized by performing in period pieces inspired by the Mexican Revolution. This cycle begins with "La Escondida" (1955). In this film, as well as in stories like "Canasta de cuentos mexicanos" (1955) and "Café Colón" (1958), she worked again with Pedro Armendáriz. In 1956 she starred in "Tizoc" with the actor and singer Pedro Infante. However, the film was not liked by the actress, despite her international success. Eventually she filmed "Beyond All Limits" (1957) with Jack Palance, and the melodramas "Ash Wednesday" and "The Empty Star" (both 1958). In 1959 she played the title role in "The Soldiers of Pancho Villa" with top billing lead Dolores del Río. In 1959 she performed in the Spanish-Mexican co-production "Sonatas" directed by Juan Antonio Bardem and the French-Mexican co-production "La Fièvre Monte à El Pao", directed by Luis Buñuel.


title: María Félix

text: In the 1960s Félix's presence in the cinema was limited to only a few films. The most prominent were "Juana Gallo" (1960), "La bandida" (1963), "Amor y sexo" ("Sapho '63", 1963)), and "La Valentina" (1966). In 1970 she filmed "La Generala", which would be her last film. The Mexican historical telenovela "La Constitución" (1971) would be her last professional acting job. Félix attempted to return to the cinema twice. First, in 1982, with the film "Toña Machetes", and again in 1986 with the film "Insólito resplandor". Neither project crystallized, and Félix never reappeared in film.


title: María Félix

text: Almost from the beginning of her career, Félix received job offers in Hollywood, but Félix herself said: "They only give me huehuenche (Indian) roles". While she was in France, Hollywood offered her the female role of "Duel in the Sun", but she passed and the part went to Jennifer Jones. Another proposal was "The Barefoot Contessa" with Humphrey Bogart, but she refused it, instead making "La Belle Otero" in France. Ava Gardner ended up with the part. Later, the director Robert Aldrich sent her the script of "The Legend of Lylah Clare", but she did not reach an agreement with the director and Kim Novak got the role.


title: María Félix

text: Félix was married four times. Her first marriage (1931–1938) was with the cosmetics sales agent Enrique Álvarez Alatorre. He fathered her only child, actor Enrique Álvarez Félix. According to journalist Sergio Almázon, once she found her son wearing a white dress and necklace. Infuriated, she beat him unconscious. Álvarez retaliated by taking their son to Guadalajara. Years later Félix was able to get her son back with the help of her second husband, Agustín Lara.


title: María Félix

text: Félix's relationship with her son was distant in his early years because she sent him to schools abroad "to discipline" him. Years later, Enrique returned to Mexico and started a career as an actor in film and television. Regarding her son, Félix said: "Enrique is a very gifted man, with admirable common sense. He's my best friend. I have so much fun with him. He's not a 'mama's boy' as many believe. Self-employed, fight like being independent. He has his own career, his audience, his poster and assumes his responsibilities without relying on me." The unexpected death of Enrique from a heart attack in 1996 affected Félix greatly.


title: María Félix

text: There are conflicting rumors that, in between her marriages to Enrique Álvarez and Agustín Lara, Félix had a brief marriage to Raúl Prado, member of the Trío Calaveras, but no concrete evidence has emerged that the marriage ever took place. Most of the rumors agree in claiming that Prado and Félix allegedly got married in 1943 after meeting on the set of "The Rock of Souls" and separated two months after the wedding. Mexican writer Enrique Serna interviewed the last survivor of the trio, Miguel Bermego, who told him he was a witness to the wedding. Prado's niece María Escalera also corroborated the marriage, saying that the marriage ended when Agustín Lara sent Félix a white piano as a gift, and "Raúl threw her out — with the piano. They later divorced. So, it appears, María Félix was my "tía política" [aunt-in-law]." In his book about Félix "María Félix: 47 pasos por el cine", Paco Ignacio Taibo I wrote about the marriage as factual, saying "While filming this movie, María would meet Raúl Prado, a member of the Los Calaveras trio. After a quick courtship they got married and shortly after they divorced. This is one of the most carefully protected chapters of the star's life. María's friends keep claiming that it was crazy." Prado's obituary in the Spanish newspaper "El País" in April 1989 reported the marriage as factual, as did some obituaries of Félix's, such as the one in Argentine newspaper "La Nación", the latter which only cited "a biography" as a source. It was cited by "TVyNovelas" as one of the "myths" surrounding Félix. It is known that Félix vehemently denied ever having married Prado. Serna, who also interviewed Félix, said "On our first meeting I asked her about her marriage to Raúl Prado (which she denied until the end), the member of the Calaveras trio who sang in Jorge Negrete's films, and she told me that if I was going to go on with those lies, we were not going to be able to work on the book." Serna added, "At one point she told me: 'Write what you want about me, because they believe anything about me. Say that I slept with an octopus.'"


title: María Félix

text: Her second marriage (1945–1947) was to the composer Agustín Lara. Félix was a fan of Lara since her adolescence. They were formally introduced by a mutual friend, the actor Tito Novaro. The couple began a highly publicized relationship, which culminated in marriage in 1945. Lara immortalized Félix in a number of songs, such as ""Humo en los ojos"" ("Smoke in the eyes"), ""Cuando vuelvas"" ("When you come back"), ""Dos puñales"" ("Two daggers"), ""Madrid"" and especially the famous theme ""María Bonita"", composed in Acapulco during their honeymoon. ""María Bonita"" would become one of Lara's most popular songs. However, the relationship ended in 1947 due to Lara's jealousy. Félix said that Lara even tried to kill her in a fit of violent jealousy.


title: María Félix

text: After her second divorce, Félix had romances with some well-known men, such as Mexican aviation entrepreneur Jorge Pasquel, Spanish bullfighter Luis Miguel Dominguín, and Argentine actor Carlos Thompson. Félix met Thompson in Argentina in 1952 during the filming of Luis César Amadori's "La pasión desnuda", and the relationship was reportedly serious to the point that they became engaged to be married, and Félix even called for her son Enrique Álvarez Félix to meet his potential stepfather. However, during filming, Félix received a call from director Emilio Fernández to offer her the leading role in "The Rapture", which Félix accepted, telling Thompson that after filming of "La pasión desnuda" ended she would return to Mexico to promote the film and announce their future marriage. However, once in Mexico, Félix canceled the wedding days before it would take place, stating that she concluded that the only thing that united her to Thompson was a mere physical attraction and not true love.


title: María Félix

text: During her stay in France, Félix was introduced by the painter Leonor Fini to French writer Jean Cau, assistant of the writer Jean Paul Sartre. Félix had a brief romance with Cau. From the end of 1950 to the spring of 1954, Félix also had a passionate same-sex affair with Suzanne Baulé, better-known as Frede, who at the time ran the cabaret "Le Carroll's" on Rue de Ponthieu in Paris, and the two women lived together at the Hotel George-V. The relationship was captured in a painting made by Fini of a plant with two flowers; one had the face of Félix and the other Frede's. Frede followed Félix on her filming trips to Buenos Aires and São Paulo upon learning of Félix's relationship with Carlos Thompson. Félix and Frede's relationship was interrupted by Félix's marriage to Jorge Negrete, but upon Negrete's death in 1953, Félix returned to Paris to briefly rekindle her relationship with Frede. However, they would violently break-up for good in 1954, leading to a trial in which Félix wanted to take back jewellery she had given to Frede and accused her of theft. Félix lost her lawsuit, and Frede was acquitted and kept the jewellery. Despite this, Félix held ownership of the painting Fini made of the two women, and afterwards Félix asked Fini to modify it to erase Frede's face, changing Frede's face to hers, resulting in the painting's two flowers both having Felix's face.


title: María Félix

text: In 1953, when Félix returned to Mexico after her stay in Europe and Argentina, she was reunited with an "old enemy": the actor and singer Jorge Negrete. Unlike their difficult first meeting ten years ago on the set of "El peñón de las ánimas", Félix found Negrete, in her own words: ""surrendered to my feet"". After a brief romance, the couple married in 1953, transmitted by radio to all Latin America, held at the House of Catipoato in Tlalpan. Negrete was already ill when the marriage took place, and died eleven months later at a hospital in Los Angeles, United States, while Félix was in Europe shooting "La Belle Otero". Félix's appearance at his funeral, dressed in trousers, caused a huge scandal, which led Félix to take refuge in Europe.


title: María Félix

text: Her fourth marriage (1956–1974), was with the Romanian-born French banker Alexander Berger. Félix met Berger in the 1940s, when they were both married. Years later they met again. Félix was married to Berger for 18 years. She tried becoming a mother again, but an accident during a filming in 1957 caused Félix to lose the child. It was at this time that she built her famous home, "La Casa de las Tortugas" (The House of Turtles), designed by Pepe Mendoza and resembling an Italian villa, in Cuernavaca. Berger died in 1974 of lung cancer months after the death of Félix's mother, which plunged her into a deep depression. To overcome this depression, she developed a new passion: horses. Some of her horses won major international equestrian awards. Félix kept her horses for 11 years.


title: María Félix

text: Her last romantic relationship was the Russian-French painter Antoine Tzapoff. About him, Félix said: "I don't know if he's the man who has most loved me, but he's who has loved me better."


title: María Félix

text: The press speculated about a strong rivalry between Félix and Dolores del Río, the other leading female figure of Mexican cinema and a successful Hollywood star. Regarding this "rivalry", Félix said: "With Dolores I had no rivalry. On the contrary, we were friends and always treated each other with great respect, each with our own personality. We were completely different. She was refined, interesting, gentle on the deal, and I'm energetic, arrogant and bossy."


title: María Félix

text: During her life, Félix was a model for many renowned painters. In her adolescence she was painted in Guadalajara by José Clemente Orozco, but Félix said, "He painted me like a skull with make-up." Félix met the artist Diego Rivera during the filming of "Río Escondido" (1947). Félix developed a friendship with him and his wife, Frida Kahlo. In 1949, Rivera painted a portrait of her, which Félix classified as "very bad". In later years, Rivera asked to borrow the painting to be displayed as part of a retrospective on Rivera's work, but Félix refused. Félix did not like the painting and she sold it to Juan Gabriel for 15 million pesos. Félix would say, “I've never liked Rivera's painting, I told him. I wanted him to paint me as a Tehuana, but he said it was very vulgar, so he painted me as he wanted ... naked, because he was very much in love with me. Later, Diego wanted that portrait for an exhibition in the Palacio de Bellas Artes, but since I did not lend it to him, he stopped talking to me for more than a year. One day there was a bricklayer fixing my house, and I sent him to paint [my nude portrait] with white to cover a bit of everything and that's how it ended up. I sold it, very badly sold by the way."


title: María Félix

text: Other renowned artists who recorded Félix in their canvases were the Surrealists Leonor Fini, Leonora Carrington, Remedios Varo and Bridget Tichenor among others. In the 1980s, the Russian-French artist Antoine Tzapoff (Félix's last partner) captured her in numerous portraits.


title: María Félix

text: Félix also had dealings with numerous Mexican and foreign writers and playwrights some of whom have dedicated several novels. Among them they are Renato Leduc, Xavier Villaurrutia, Salvador Novo, Pita Amor, Jean Cocteau and Octavio Paz. Luis Spota was inspired by her life for his novel "The Empty Star", and Félix even starred in the film version in 1958. Carlos Fuentes did the same in the novel "Zona sagrada". Her relationship with Fuentes was terminated when the author made the play "Orchids in the Moonlight", in which he parodies the figures of Félix and Dolores del Río. Félix, angry, called him ""mujerujo"" ("womanish").


title: María Félix

text: Songs were composed for Félix, including "María de Todas las Marías" by Juan Gabriel and "Je l'aime à mourir" by Francis Cabrel.


title: María Félix

text: In fashion, Félix was dressed by designers like Christian Dior, Givenchy, Yves Saint Laurent, Chanel, and Balenciaga. The House of Hermès (Couture Department) designed extravagant creations just for her. She was a noted collector of fine antiques, favoring pieces like her famous collection of Second French Empire furniture. She was also a jewelry connoisseur and had an extensive jewelry collection, including the 41.37 carat (8.274 g), D-flawless "Ashoka" diamond. In 1968, she commissioned a serpent diamond necklace from Cartier. The result was a completely articulated serpent made out of platinum and white gold and encrusted with 178.21 carats (35.642 g) of diamonds. In 1975, she again asked Cartier to create a necklace for her, this time in the shape of two crocodiles. The two crocodile bodies were made of 524.9 grams of gold, one covered with 1,023 yellow diamonds, while the other was adorned with 1,060 circular cut emeralds. Later she sold most of her jewelry back to Cartier. She also left a Rolls-Royce in Paris.


title: María Félix

text: Since Félix's death, these jewelry pieces have been displayed as part of "The Art of Cartier Collection" in several museums around the world. To pay tribute to the actress, in 2006 Cartier debuted its "La Doña de Cartier collection". "The La Doña de Cartier watch" with reptilian links was created to impress by its wild look. The case of the "La Doña de Cartier" features a trapezoid shape with an asymmetrical profile reminiscent of a crocodile's head. The wristband of the watch resembles the contours of a crocodile in large, bold and gold scales. The "La Doña de Cartier Collection" also includes jewelry, accessories, and handbags.


title: María Félix

text: María Félix died in her sleep on 8 April 2002, her 88th birthday in Mexico City. She was buried in her family's mausoleum alongside her son Enrique and parents at the Panteón Francés located in Mexico City. In 2018, Google celebrated Felix's 104th birthday with a Google Doodle. A skeletal version of Felix appears briefly in the 2017 Pixar film "Coco", as a guest at a party in the Land of the Dead, with luchador El Santo as her date.


title: María Félix

text: On 21 July 2022, the TelevisaUnivision produced biopic "", starring Sandra Echeverría, was released on streaming service Vix.


title: Susan Alice Buffett

text: Susan Alice Buffett (born July 30, 1953) is an American philanthropist who is the daughter of Warren Buffett and Susan Thompson Buffett. Her charitable work has focused largely on the Sherwood Foundation, formerly known as the Susan A. Buffett Foundation, an organization in Omaha that provides grants in public education, human services and social justice in the interest of promoting the welfare of children from lower-income families. She is also on the boards of the Susan Thompson Buffett Foundation, The Buffett Early Childhood Institute, and Girls, Inc. According to a 2010 interview with her brother Howard Graham Buffett, Buffett's philanthropic focus has consistently remained on children, education and family issues, but she has also committed to other causes, including Debt, AIDS, Trade, Africa, a non-governmental organization dedicated to various improvements in Africa.


title: Susan Alice Buffett

text: She attended the University of Nebraska–Lincoln, where she majored in home economics, and studied at the University of California, Irvine, where she majored in social ecology.


title: Susan Alice Buffett

text: Her parents separated in the late 1970s (though they remained married until her mother's death in 2004). Despite his wealth, Warren Buffett encouraged his children to be financially independent; Susan Buffett recalled in 2006 that in spite of her father's generosity, he once refused her a personal loan of $41,000 to expand her kitchen. Her foundation, however, was funded primarily by $1 billion in shares from her father, Warren Buffett.


title: Susan Alice Buffett

text: In 1983, Buffett wed Allen Greenberg, a lawyer for Public Citizen, whom she had met in Washington. The couple divorced in 1995.


title: Susan Alice Buffett

text: In 1987, Greenberg became the first director of the Buffett Foundation, a title he retained after the couple's divorce.


title: Alfredo Morelos

text: Alfredo José Morelos Aviléz (born 21 June 1996) is a Colombian professional footballer who plays as a striker for Scottish Premiership club Rangers and the Colombia national team.


title: Alfredo Morelos

text: Morelos began his senior career with Deportivo Independiente Medellín in his homeland before moving to Finland to play for Helsinki side HJK, initially on loan then permanently. In June 2017, he moved to Scotland to join Rangers, for a reported fee of £1m. Morelos currently holds the club record for most goals scored in European competitions with 29 goals.


title: Alfredo Morelos

text: After making his way through the Colombian youth system representing Colombia U17 and Colombia U20 with whom he reached the final of the 2015 South American U-20 Championship, Morelos made his senior international debut for Colombia in 2018 and was part of the squad that finished third at the 2021 Copa América.


title: Alfredo Morelos

text: Born in Cereté, Colombia, Morelos lived with his sisters, his parents and his grandmother, but after joining Independiente Medellín, he had to live away from his family. Morelos said in an interview with AS.com that he is married. While in Finland, Morelos said though it was tough to adapt to the country, as his family decided to stay in Colombia and he had to travel to South America to be with them. Morelos also revealed that it was his father who got him into football and goalscoring.


title: Alfredo Morelos

text: Morelos began his career at Independiente Medellín in 2011 at the age of 15 and worked his way through the academy ranks he was promoted to the first team by then manager Hernán Darío Gómez, he made his Independiente Medellín debut aged 18 years and 22 days, in a 2–2 draw with Rionegro Águilas in the Copa Colombia group stage on 13 July 2014 coming on as a 74th-minute substitute. On 24 July 2014 Morelos started his first game for Independiente Medellín and scored his first ever senior goal aged 18 years, 1 month and 3 days in a 2–0 win over Evigado in the Copa Colombia. 7 days later on 31 July 2014 Morelos scored a brace in two minutes in a 3–2 win over Jaguares de Córdoba during a Copa Colombia group B match taking his tally to 3 goals in 3 games. Morelos featured three more times for Independiente Medellín in the group stage of the competition scoring a further two goals, one against Leones on 21 August 2014 in a 3–0 win and one 7 days later against Evigado in a 1–1 draw. Independiente Medellín finished second in their group with 21 points, level on points with first placed Atlético Nacional and progressed through to the last 16 of the 2014 Copa Colombia but where defeated 5–2 on aggregate by Deportes Tolima, Morelos made an appearance in both legs of the tie.


title: Alfredo Morelos

text: Morelos made his Categoría Primera A debut for Independiente Medellín on 31 August 2014 in a 3–0 win over Águilas Doradas coming on as an 82nd-minute substitute. 6 days later on 6 September 2014, Morelos scored his first Categoría Primera A goal for Independiente Medellín during injury time against Fortaleza Esporte Clube to win the match 3–2 in the 92nd minute. Morelos made a further four appearances in the 2014 Categoría Primera A but the goal against Fortaleza would be his last of the season.


title: Alfredo Morelos

text: Independiente Medellín finished runners-up to Independiente Santa Fe in the 2014 Finalización after losing the final 3–2 on aggregate, Morelos made an appearance in the 2nd leg of the tie on 21 December 2014 coming on as a 66th-minute substitute in a 1–1 draw.


title: Alfredo Morelos

text: Morelos ended the 2014 season with fourteen appearances and six goals in all competitions for Independiente Medellín and received a 2014 Categoría Primera A Finalización runner-up medal.


title: Alfredo Morelos

text: Morelos made eleven appearances in all competitions in the 2015 season, six in the Categoría Primera A and five in the Copa Colombia but failed to find the net in any of the matches. Independiente Medellín reached the final of the Categoría Primera A 2015 Apertura but were defeated 2–1 on aggregate by Deportivo Cali with Morelos coming as a 68th-minute substitute in the 2nd leg on 8 June 2015 which ended 1-1, earning Morelos his second runner-up medal.


title: Alfredo Morelos

text: In January 2016 it was reported in the Colombian media that Morelos was linked with a move away from Independiente Medellín to European side HJK Helsinki.


title: Alfredo Morelos

text: Morelos made a total of twenty five appearances for Independiente Medellín scoring six goals in all competitions during his time with the club.


title: Alfredo Morelos

text: On 5 February 2016, HJK announced the signing of Morelos on loan, with an option of buying the striker on a permanent basis.


title: Alfredo Morelos

text: Morelos made an immediate impact on his debut four days later in the Finnish League Cup, coming on as a substitute in the Stadin derby against HIFK and scored in the 90th minute to make the game 3–3. A week later on 18 February 2016, Morelos scored a brace in the League Cup once again, in a 2–2 draw against Inter Turku. Morelos also made an impact in the league and started the season well scoring six goals in the first five matches against VPS, RoPS (twice), SJK and AC Kajaani (twice). Morelos would then not score for six matches before scoring against rivals HIFK in a 2–1 win on 26 May 2016, Morelos scored his first and only hat–trick for HJK and the first of his career aged 19 years, 11 months and 25 days in the Finnish Cup, in a 4–1 win over KuPS on 15 June 2016, he followed this up by scoring in a 3–2 win over SJK in the league three days later. Five days later, on 23 June 2016, he scored in the semi–final of the Finnish Cup, in a 3–0 win over Lahti to send them through to the final.


title: Alfredo Morelos

text: In the UEFA Europa League Qualifiers, Morelos scored four goals in six UEFA Europa League appearances against Atlantas (twice), Beroe Stara Zagora and IFK Göteborg all of which were in the first leg of each tie, HJK where deafted 3–2 on aggregate by IFK Göteborg after a 2–0 defeat away in the Third qualifying round and exited the competition. He then scored four more league goals by August against PK-35 Vantaa, FC Lahti, KuPS (twice) and scored against KuPS for the third time in another encounter weeks later on 28 August 2016. After scoring five goals in six league matches against PS Kemi, Lahti, RoPS, HIFK, and Mariehamn and scoring HJK's only goal in the final of the Finnish Cup against SJK which ended 1-1 after full-time (though the club lost 7–6 in the penalty shoot–out with Morlelos scoring his penalty), Morelos was named September Player of the Month. Morelos scored in the second last game of the season against title challengers, IFK Mariehamn, on 14 October 2016 which ended in a 1–1 draw. However, he was unable to help the club win the league after IFK Mariehamn won it by three points in the last game of the season after HJK drew 0–0 with SJK and IFK Mariehamn defeated Ilves 2–1.


title: Alfredo Morelos

text: At the end of the 2016 season, Morelos finished HJK's top goal scorer in the league with sixteen goals in thirty appearances and top goal scorer in all competitions with thirty goals in forty three appearances, he was also runner-up top goalscorer in the league missing out on the top spot by one goal to Roope Riski who finished the season with seventeen goals. Morelos earned a Veikkausliiga runner-up medal, a Finnish Cup runner-up medal and was awarded the league's best striker of the year award.


title: Alfredo Morelos

text: HJK took up their option to sign Morelos permanently and he started 2017 by scoring in the first game of the season in a 3–2 victory over FC Honka on 28 January 2017 in the Finnish Cup group stage. He followed this up by scoring three goals in the remaining four group games against KTP, GrlFK, and HIFK. The latter being in a 4–0 win over HJK's Derby rivals on 1 March 2017, which was HJK's biggest win over HIFK since 1984 when they beat HIFK 9–1 in the Finnish Cup. Morelos helped HJK finish top of their group with five wins out of five and progress through to the Quarter-final. On 1 April 2017 Morelos scored the only goal in the 75th minute in a 1–0 victory over Ilves in the Finnish Cup Semi-final to send HJK into the final and take his tally up to five goals in seven cup matches.


title: Alfredo Morelos

text: Morelos got off the mark in his first 2017 Veikkausliiga league match on 5 April 2017 when he scored in a 5–0 win over VPS, five days later Morelos scored in a 2–0 victory over KuPS. Between 23 April 2017 and 19 May 2017 Morelos would score six goals over five consecutive matches against JJK, Inter Turku, VPS, Ilves (twice), and PS Kemi. Morelos would not score in HJK's next two games which both ended as 0-0 draws against HIFK and Lahti, then scored a brace on 31 May 2017 in a 6–0 win over SJK. For his excellent form and goalscoring which included ten goals in eleven appearances Morelos was awarded the Veikkausliiga Player of the month for both April 2017 and May 2017. Morelos scored his final goal for HJK in his last match for the club before his transfer to Rangers on 4 June 2017 in a 4–1 victory over RoPS.


title: Alfredo Morelos

text: Despite leaving halfway through the Finnish season Morelos finished as HJK's second top goalscorer for the 2017 season after he scored eleven league goals in twelve appearances, five cup goals in seven appearances and sixteen goals over nineteen appearances in all competitions and was joint-seventh top goalscorer in the 2017 Veikkausliiga. HJK went on to win the 2017 Veikkausliiga by twenty points and the 2016–17 Finnish Cup defeating SJK 1–0 in the final on 23 September 2017 with Morelos receiving two winners medals for his contributions over the season.


title: Alfredo Morelos

text: Morelos made a total of sixty two appearances for HJK scoring forty six goals, he won three player of the month awards, one striker of the year award and two trophies during his time at the club.


title: Alfredo Morelos

text: Morelos signed a three-year deal with Rangers on 19 June 2017 for an undisclosed fee, which was widely reported to be £1 million.


title: Alfredo Morelos

text: He made his debut for Rangers in a 1–0 home win against Progrès Niederkorn in the Europa League first qualifying round first leg ten days later, coming on in the 77th minute in place of Martyn Waghorn. On 9 August, Morelos scored his first two Rangers goals in a 6–0 win over Dunfermline Athletic also at Ibrox in the second round of the Scottish League Cup. Three days later he scored his first goal in the Scottish Premiership, heading the team into an early lead in a 3–2 home loss to Hibernian. He quickly became a favourite with Rangers supporters, who nicknamed him "El Bufalo".


title: Alfredo Morelos

text: He finished the season with 18 goals, tied (with Josh Windass) as the club's top scorer for the season; 14 of those strikes were in the league, the second-highest total behind Kris Boyd of Kilmarnock.


title: Alfredo Morelos

text: Rangers started the season playing in the UEFA Europa League qualifying rounds, and goals against NK Osijek and NK Maribor from Morelos helped Rangers progress to the group stages of the tournament. However, a sign of future disciplinary issues arose when he was sent off against Aberdeen at Pittodrie Stadium in the opening league fixture for a kick out at an opponent, although the red card was rescinded on appeal. He continued to score goals early in the season; scoring in 2–0 win over St Mirren in the league, and netting a hat trick in a 3–1 win over Kilmarnock in the League Cup. In September 2018 he signed a new contract with Rangers, until 2022. After several games without a goal, he resumed his goalscoring form on 26 September 2018 when he netted Rangers second goal in a 5–1 win over St Johnstone. Morelos then went on to score 10 goals in the next 13 games, including a brace in a 3–1 win over Rapid Vienna in the Europa League.


title: Alfredo Morelos

text: On 6 February 2019, Morelos scored twice in a 4–2 victory away at Aberdeen but also received his fourth red card of the season and his third against the same rival opposition (one was later rescinded). Manager Steven Gerrard stated that he was trying to help the player, and the club appealed the latest red card from the Aberdeen game. The appeal was unsuccessful, but Morelos was able to play in a Scottish Cup tie while serving a three-match League ban, and scored four goals in that match, a 5–0 win over Kilmarnock, to take his tally for the 2018–19 campaign to 27 in 39 appearances.


title: Alfredo Morelos

text: In March 2019 he signed a new contract with Rangers, his second of the 2018–19 season, until 2023. Later that month, Morelos received a straight red card for an incident with Celtic's Scott Brown in an eventual 1–2 loss for Rangers in the Old Firm derby; it was his fifth sending off of the season, although one of them was rescinded. Following the incident, manager Steven Gerrard said he could no longer defend the player. Despite then serving a four-match suspension, Morelos finished as the top goalscorer in the 2018–19 Scottish Premiership with 18 league goals. His goal against Kilmarnock in the last league match of the season was his 100th competitive goal at club level.


title: Alfredo Morelos

text: During the summer of 2019 Morelos turned down a lucrative contract offer from Chinese Super League side Hebei China Fortune worth £30 million that would have seen the Colombian net £10 million per year before tax, the fee believed to be offered to Rangers was over £15 million although no official bid was received. Morelos started the 2019–20 season by scoring a goal against St Joseph's F.C. in the first leg of their Europa League First Qualifying round tie in Gibraltar. He followed this up in the return leg at Ibrox, by scoring his third hat-trick for Rangers. He then scored a goal and made two assists against FC Midtjylland in the Europa League Third Qualifying round. Three days later he came on as a substitute on the 76th minute during Rangers' first home league game of the season against Hibernian and scored a brace in a 6–1 victory, taking his tally to 7 goals in 7 games for the season.


title: Alfredo Morelos

text: In October 2019, Morelos was allegedly racially abused by Hearts fans after he scored during a match between the two sides. At the end of November, he scored twice in a 2–2 draw away at Feyenoord in the Europa League. In scoring these goals, Morelos became the first Rangers player to score in four consecutive matches in European competition. After the game, manager Gerrard praised Morelos and highlighted his improved disciplinary record over the season to date.


title: Alfredo Morelos

text: Over the course of a week in December, Morelos failed to score from several chances (including a penalty) as Rangers lost the 2019 Scottish League Cup Final 1–0 to Celtic, struck his side's goal in a 1–1 draw with BSC Young Boys which ensured qualification for the 2019–20 UEFA Europa League knockout phase, and netted the winner in a league match away to Motherwell, but was then sent off for a second booking after making gestures to the home supporters during his celebrations. Morelos, however, found himself again the centre of controversy at the end of Rangers' 2-1 league win over Celtic at Parkhead on 29 December. He was sent off for diving, and made some gestures with his hand across his throat as he left the field. Rangers later claimed he had been subjected to racist abuse from Celtic supporters at the game.


title: Alfredo Morelos

text: By the time of the mid-season break at the end of December, Morelos had scored 28 goals in 35 games. He also set a UEFA individual record for most goals in a European campaign before Christmas, with 14.


title: Alfredo Morelos

text: Despite much speculation that Morelos would leave Rangers and a £16 million bid from Lille which if accepted would have been a record transfer fee received by Rangers, he remained with the club for the 2020–21 season. He started the season in goalscoring form, netting five in his first seven games including braces against St Mirren in the Scottish Premiership and Lincoln Red Imps during Europa League qualifying. After scoring his fifth goal of the season against Hibernian on 20 September 2020, Morelos would not find the net again in domestic competitions until 30 December when he scored against St Mirren. He continued his rich vein of form in the group stage of the Europa League by scoring against both Polish side Lech Poznań on 29 October 2020, and Portuguese giants Benfica on 5 November. By scoring the latter goal at the Estádio da Luz, Morelos surpassed Ally McCoist as Rangers top goalscorer in European competitions, holding the club record with 22 goals.


title: Alfredo Morelos

text: Morelos scored his first goals of 2021, netting a brace on 10 January during a 2–1 league win away to Aberdeen. On 27 January 2021, he played and scored the only goal in a 1–0 Premiership win over Hibernian at Easter Road. His performance proved controversial due to an off the ball incident where he appeared to stamp on Hibernian player Ryan Porteous, and was subsequently given a three match ban. On 21 February, he scored a bizarre goal during a league match against Dundee United as a clearance from United's goalkeeper Benjamin Siegrist rebounded off Morelos's body into the goal. He followed this up four days later with a goal in the Europa League Round of 32 match against Royal Antwerp. On 21 March, he scored his first goal in the Old Firm derby during a 1–1 draw away at Celtic Park; later reports emerged that he was the subject of racial abuse over social media perpetrated by alleged opposition fans. With three goals in three appearances, Morelos ended March 2021 by being named Scottish Premiership Player of the Month. On 2 May, Morelos scored against Celtic again in a 4–1 win at Ibrox, which was a biggest margin of victory against their rivals in 14 years and ensured that Rangers stayed unbeaten against Celtic in all league matches for the first time since the 1999–2000 season.


title: Alfredo Morelos

text: This would be Morelos's final goal of the season but he played a part in the final two fixtures, a 3–0 away win against Livingston on 12 May where he provided an assist and a 4–0 home win against Aberdeen on 15 May which meant that Rangers completed the full Scottish Premiership season unbeaten in addition to winning the top tier title for the first time in 10 years and 55th time overall. Morelos won his first trophy after four seasons with the club. He ended the season with 17 goals in 44 appearances in all competitions.


title: Alfredo Morelos

text: Again there was much speculation over the summer transfer window that Morelos would leave Rangers; the club rejected a £10 million bid from Porto (against whom Morelos had scored twice in the previous year's Europa League group stage). Morelos stayed a Rangers player but missed the first two matches of the season after being called up to the Colombia national side for the 2021 Copa América. On 10 August 2021, he scored his first goal of the season and his first ever UEFA Champions League goal in the second leg of Rangers Third qualifying round match against Malmö FF. He became the first Rangers player to score a Champions League goal for ten years – the last also came against Malmö; despite this Rangers lost the tie 4–2 on aggregate and dropped into the UEFA Europa League Play-off. On 19 August, Morelos scored the only goal over two legs against Alashkert in the Europa League Play-off round to help Rangers reach the Europa League group stage for the fourth consecutive season. Three days later Morelos scored his first league goal of the season in a 4–2 away win over Ross County, a game in which he also recorded an assist.


title: Alfredo Morelos

text: Morelos would not score again for a month until a 2–0 win over Livingston on 22 September 2021 at Ibrox in the quarter-final of the Scottish League Cup. A few weeks later on 3 October he scored the winning goal in a 2–1 league win over Hibernian. On 24 October 2021 he again scored the winner against St Mirren as Rangers came behind to score twice in as many minutes and win 2–1. In doing so Morelos scored his 100th goal for Rangers in his 199th match for the club, he celebrated by jumping into the crowd which he received a yellow card for. He became the first player to score 100 goals for Rangers in over five years and the 18th post-war player to do so, with Kenny Miller being the last player to achieve this in 2016. Three days later Morelos marked his 200th appearance for the club by scoring his 101st goal against Aberdeen in a 2–2 draw at Ibrox.


title: Alfredo Morelos

text: On 25 November 2021, in their first match under new manager Giovanni van Bronckhorst, Rangers defeated Sparta Prague 2–0 in the Europa League group stage at Ibrox; Morelos, who had not scored in the previous four group matches, got both goals to ensure their head-to-head record was superior to their opponents (who had won 1–0 in the Czech Republic) and secured qualification for the knockout stages of the competition for the third season in a row. Under van Bronckhorst Morelos looked like a rejuvenated player and hit a rich vein of form in the month of December scoring a further 4 goals in 5 league matches against Dundee, Hearts, St Johnstone and St Mirren. The latter of which broke another Rangers record as Morelos scored his 107th goal for the club and became the all-time overseas top goalscorer, a record which was previously held by South African left winger Johnny Hubbard since 1959. For his form and goalscoring Morelos ended the year on a high by winning the player of the month award for December 2021.


title: Alfredo Morelos

text: Morelos scored his first goals of 2022 by netting a brace against Hearts on 6 February at Ibrox during a 5–0 win, a game in which he also recorded an assist. The two goals took Morelos league goals tally up to 10 for the season and he became the first player to score double figures in the Scottish Premiership for Rangers for 5 consecutive seasons since Kris Boyd between 2005/06 and 2009/10. Three days later Morelos scored in a 2–0 home win over Hibernian. On 17 February, Morelos scored a goal and caused an own goal with his shot deflecting off of Dan-Axel Zagadou as Rangers recorded a historic 4–2 European away win over German giants and favourites to win the tournament Borussia Dortmund at the Westfalenstadion in the first leg of the 2021–22 UEFA Europa League knockout round play-offs. Rangers progressed through to the last 16 of the Europa League following a 6-4 aggregate win over Dortmund and were drawn against Serbian side Red Star Belgrade, in the first leg at Ibrox on 10 March Morelos scored in a 3–0 win. Morelos season was cut short with his final appearance being in a 2–1 away win over Dundee on 20 March after he sustained an injury whilst on International duty which required him to undergo surgery on his thigh. On 5 April Rangers put out an official statement on their website to confirm the surgery was a success but Morelos would be out for the rest of the season.


title: Alfredo Morelos

text: Despite his injury Morelos would finish the season as Rangers joint-top goalscorer with 18 goals (tied with James Tavernier) in 42 appearances and received the award for Rangers Player Of The Year, he was also included in the 2021-22 PFA Scotland Team of the Year. Without Morelos Rangers reached the Europa League Final which they narrowly lost to Eintracht Frankfurt 5–4 on penalties at the Ramón Sánchez Pizjuán Stadium in Seville after the two sides could not be separated after 120 minutes. Although Morelos never took part in the final he received a runners-up medal for his earlier contributions in the European campaign which included vital goals for Rangers to be able progress to that stage in the tournament. Before the match Morelos was spotted in the streets of Seville taking pictures and interacting with fans who had travelled over in large numbers for the occasion.


title: Alfredo Morelos

text: After almost five months out of action Morelos made his return on 6 August 2022 in a league game against newly promoted side Kilmarnock at Ibrox coming as a 64th minute substitute for Antonio Čolak, 14 minutes later Morelos scored his first goal of the season as Rangers won the match 2-0. On 20 August 2022 during a 2-2 draw with Hibs, Morelos received his eighth red card for Rangers after he swung his arm in the face of Marijan Čabraja when both players went to header the ball. The commentators and pundits felt the red card was harsh and it meant that Rangers had to finish the game with nine men as John Lundstram had just been sent off 9 minutes prior to the incident. Following this manager Giovanni van Bronckhorst took the decision to drop Morelos from the squad to face PSV Eindhoven in the Champions League Play-off round (which Rangers went on to win 3-2 on aggregate and qualify for the Champions League group stage for the first time in 12 years) citing concerns over his attitude and fitness levels but stressed that Morelos still had a future at the club if he could prove his commitment. Morelos made his first start for Rangers in six months and his first ever Champions League group stage appearance on 14 September 2022 in a 3-0 defeat against S.S.C. Napoli at Ibrox. On 1 October 2022 Morelos scored his second goal of the season, 13 minutes after coming on as a 63rd minute substitute in a 4-0 win over Hearts at Tynecastle. On 29 October 2022 Morelos again scored after coming on as substitute in a 4-1 victory over Aberdeen at Ibrox, the goal took his tally up to 115 for Rangers and meant that he became the 30th all-time top goalscorer for the club (level with Rangers legend Mark Hateley).


title: Alfredo Morelos

text: Morelos was called up by Colombia U17 for the first time in July 2012 and made his Colombia U17 debut in a 9–0 win against Paraguay U17 on 2 April 2013. He went on to make four appearances for the U17 side. In September 2014, Morelos was called up by Colombia U20 for the first time and made his Colombia U20 debut on 16 January 2015, in a 6–0 win against Uruguay. Morelos was in the Colombia squad when they finished runner's-up in the 2015 South American Youth Football Championship.


title: Alfredo Morelos

text: On 28 August 2018, Morelos received his first call-up to the Colombian senior national team for the friendly matches against Venezuela and Argentina. He made his international debut on 8 September 2018, coming on as a 77th-minute substitute against Venezuela. In the last minute he provided an assist for Yimmi Chará's winning goal for Colombia. He made a further two appearances in March 2019 in friendlies against Japan and South Korea, making his first international start against the latter side. He was not named in the 40-man preliminary squad announced on 16 May for the 2019 Copa América, with several media outlets speculating his poor on-field discipline at club level in Scotland contributed to his omission.


title: Alfredo Morelos

text: Morelos returned to the national side in October 2019 and featured in friendlies against Chilie and Algeria. Morelos scored his first international goal on 16 November 2019, in a friendly during injury time against Peru as Colombia won 1–0.


title: Alfredo Morelos

text: On 10 October 2020, Morelos played his first ever FIFA World Cup qualification game for Colombia coming on as a substitute in the 59th minute as Colombia defeated Venezuela 3–0. He made two more appearances in the qualifiers during 2020; one against Chile on 14 October and the other against Uruguay on 13 November.


title: Alfredo Morelos

text: Morelos was selected in the 28-man Colombia squad for the 2021 Copa América which took place in June 2021, but missed the warm up World Cup qualifier games against Peru and Argentina after he tested positive for corona virus during a routine round of testing whilst on international duty. Morelos made his Copa América debut on 21 June 2021, coming on as an 81st-minute substitute in a group game against Peru which Colombia lost 2–1. Colombia progressed through their group and reached the knockout stages of the tournament but were eventually defeated in the Semi-final by Argentina 4–3 on penalties. Morelos was an unused substitute in Colombia's final game against Peru in the Third place play-off which Colombia won 3–2 after a dramatic 25 metre goal during injury time by Luis Diaz which secured Colombia Third place at the 2021 Copa América and earned Morelos a bronze medal.


title: Alfredo Morelos

text: Morelos has been with his childhood sweetheart Yesenia Herrera since 2014, the couple married in January 2016 and have a daughter together Leonela Morelos who was born in May 2020.


title: Time (2021 TV series)

text: Time is a 2021 three-part TV drama written by Jimmy McGovern, directed by Lewis Arnold, and starring Sean Bean and Stephen Graham. It was first broadcast on BBC One on 6 June 2021.


title: Time (2021 TV series)

text: Mark Cobden is newly imprisoned, consumed by guilt for his crime, and way out of his depth in the volatile world of prison life. He meets Eric McNally, an excellent prison officer doing his best to protect those in his charge. However, when one of the most dangerous inmates identifies his weakness, Eric faces an impossible choice between his principles and his family.


title: Time (2021 TV series)

text: The wings and cells prison scenes were filmed at HM Prison Shrewsbury, a former prison which was decommissioned in 2013, however most of the filming took place in the Liverpool City Region. The cameras moved to Liverpool to create the rest of the prison with a mash-up of courtrooms, police stations and education buildings, as well as exterior locations such as the Silver Jubilee Bridge in Widnes and the Southport Pier.


title: Time (2021 TV series)

text: Writing in "The Guardian", Lucy Mangan wrote: "The performances of Bean and Graham are, even though we have come to expect brilliance from them both, astonishing. So, too, are those from everyone in smaller roles, none of which is underwritten or sketchy, and who thicken the drama into something more profoundly moving and enraging at every turn". Billie Schwab Dunn, writing for "Metro", praised the show, which was "elevated by the central performances – particularly Bean, who gently grounds us and provides a beam of light in all that darkness".


title: Manchester United W.F.C.

text: Manchester United Women Football Club is a professional football club based in Leigh, Greater Manchester, England, that competes in the Women's Super League (WSL), the top tier of English women's football, after gaining promotion from the Championship at the end of the 2018–19 season.


title: Manchester United W.F.C.

text: A team called Manchester United Supporters Club Ladies began operations in the late 1970s and was unofficially recognised as the club's senior women's team. They became founding members of the North West Women's Regional Football League in 1989. The team became increasingly competitive throughout the 90s at various levels of the FA Women's National League up until 2001.


title: Manchester United W.F.C.

text: In 2001, the team formed an official partnership with Manchester United. However, the club began to stagnate with a string of successive mid-table finishes in the third division. The team were disbanded four years later, in 2005, shortly after Malcolm Glazer's completed takeover with the new regime deeming the women's team to not be part of the "core business" and unprofitable. A media spokesman for Manchester United also claimed the club wanted to focus on its women's academy instead of its senior team.


title: Manchester United W.F.C.

text: In March 2018, Manchester United announced their intentions to reintroduce a women's football team. Manchester United Women Football Club were founded on 28 May 2018, following the club's successful application to join the newly-formed 2018–19 FA Women's Championship. It marked the club's return to women's football after a thirteen-year absence; though the club's academy continued via the Manchester United Foundation, with the likes of Izzy Christiansen and Katie Zelem being produced by United's Centre of Excellence academy. Casey Stoney was appointed as the club's first head coach on 8 June, with their inaugural 21-player squad announced just over a month later.


title: Manchester United W.F.C.

text: The team's first game back was on 19 August 2018, where they won 1–0 in an away game against Liverpool in the FA Women's League Cup, with Lizzie Arnot scoring their first competitive goal in thirteen years. Three weeks later, their opening Championship encounter ended in a 12–0 victory away to Aston Villa. On 17 April 2019, United secured promotion to the FA Women's Super League following a 5–0 win against the same opposition. They clinched the FA Women's Championship title three days later following a 7–0 win at home to Crystal Palace. In May 2019, Manchester United were named FA Women's Championship Club of the Year at the 2019 FA Women’s Football Awards.


title: Manchester United W.F.C.

text: The 2019–20 season was the team's maiden FA WSL campaign. The season opener was a Manchester derby, held at the City of Manchester stadium. Manchester City won the match 1–0 in front of a then league record attendance of 31,213. United's first top-flight campaign was ended prematurely after the season was curtailed by the COVID-19 pandemic with the team awarded fourth place on a points per game basis. The last game was a 3–2 victory over Everton on 23 February 2020, with Leah Galton scoring twice and Ella Toone scoring once for Manchester United in the first game played at Everton's new Walton Hall Park stadium. In the FA Cup, Manchester United lost 3–2 to Manchester City in the fourth round, the first time they had lost in the first knockout round of a competition. However, they repeated their best League Cup finish by reaching the semi-finals for the second consecutive year, losing 1–0 to eventual cup winners Chelsea.


title: Manchester United W.F.C.

text: Manchester United Women played their first game at Old Trafford, against West Ham United, during the men's March 2021 international break. Manchester United won the game 2–0.


title: Manchester United W.F.C.

text: On 12 May 2021, Manchester United announced that Casey Stoney would stand down from her role as Head Coach at the end of the season. On 29 July 2021, Marc Skinner was announced as the new head coach on a two-year contract, with the option for a further year.


title: Manchester United W.F.C.

text: Following the club's acceptance into the 2018–19 FA Women's Championship, it was revealed that the women's team would be based in Broughton, Salford at The Cliff training ground; subject to completion of redevelopment work. In the meantime United play their fixtures at Leigh Sports Village. Moss Lane serves as a backup venue for when Leigh Sports Village is unavailable. Ewen Fields has also been used as a contingency plan, hosting United’s FA Cup fifth round match against London Bees in February 2019. In March 2021, it was announced that the women's team would play their first ever game at Old Trafford later that month against West Ham United. In March 2022, the team played for the second time at Old Trafford and the first with fans present, beating Everton 3–1 in front of a club record 20,241 fans.


title: Manchester United W.F.C.

text: Despite not having a senior women's team for many years, Manchester United have continued to run a girls regional talent club up to under-16 level in accordance with FA regulations. The club's partner charity, The Manchester United Foundation, also works in coaching girls at all ages across the Greater Manchester region. Ahead of the 2019–20 season, Manchester United entered a full-time U21 team into the FA WSL Academy League for the first time, managed by Charlotte Healy. The club's development team had contested the WSL Academy Cup final against Arsenal the previous season.


title: P-adic number

text: In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.


title: P-adic number

text: These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers. The -adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this. For example, the field of -adic analysis essentially provides an alternative form of calculus.


title: P-adic number

text: More formally, for a given prime , the field of -adic numbers is a completion of the rational numbers. The field is also given a topology derived from a metric, which is itself derived from the -adic order, an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in . This is what allows the development of calculus on , and it is the interaction of this analytic and algebraic structure that gives the -adic number systems their power and utility.


title: P-adic number

text: The in "-adic" is a variable and may be replaced with a prime (yielding, for instance, "the 2-adic numbers") or another expression representing a prime number. The "adic" of "-adic" comes from the ending found in words such as dyadic or triadic.


title: P-adic number

text: where formula_3 is an integer and each formula_4 is also an integer such that formula_5 This expansion can be computed by long division of the numerator by the denominator, which is itself based on the following theorem: If formula_6 is a rational number such that formula_7 there is an integer formula_8 such that formula_9 and formula_10 with formula_11 The decimal expansion is obtained by repeatedly applying this result to the remainder formula_12 which in the iteration assumes the role of the original rational number formula_1.


title: P-adic number

text: The -"adic expansion" of a rational number is defined similarly, but with a different division step. More precisely, given a fixed prime number formula_14, every nonzero rational number formula_1 can be uniquely written as formula_16 where formula_3 is a (possibly negative) integer, formula_18 and formula_19 are coprime integers both coprime with formula_14, and formula_19 is positive. The integer formula_3 is the -adic valuation of formula_1, denoted formula_24 and formula_25 is its -adic absolute value, denoted formula_26 (the absolute value is small when the valuation is large). The division step consists of writing


title: P-adic number

text: where formula_8 is an integer such that <math>0\le a and formula_12 is either zero, or a rational number such that formula_30 (that is, formula_31).


title: P-adic number

text: obtained by repeating indefinitely the above division step on successive remainders. In a -adic expansion, all formula_4 are integers such that formula_36 with formula_37, the process stops eventually with a zero remainder; in this case, the series is completed by trailing terms with a zero coefficient, and is the representation of formula_1 in base-.


title: P-adic number

text: The existence and the computation of the -adic expansion of a rational number results from Bézout's identity in the following way. If, as above, formula_39 and formula_19 and formula_14 are coprime, there exist integers formula_42 and formula_43 such that formula_44 So


title: P-adic number

text: The uniqueness of the division step and of the whole -adic expansion is easy: if formula_51 one has formula_52 This means formula_14 divides formula_54 Since formula_55 and formula_56 and since formula_14 divides formula_58 it must be that formula_59


title: P-adic number

text: The -adic expansion of a rational number is a series that converges to the rational number, if one applies the definition of a convergent series with the -adic absolute value.


title: P-adic number

text: In the standard -adic notation, the digits are written in the same order as in a standard base- system, namely with the powers of the base increasing to the left. This means that the production of the digits is reversed and the limit happens on the left hand side.


title: P-adic number

text: The -adic expansion of a rational number is eventually periodic. Conversely, a series formula_60 with formula_61 Bézout's identity for 5 and the denominator 3 is formula_62 (for larger examples, this can be computed with the extended Euclidean algorithm). Thus


title: P-adic number

text: For the next step, one has to "divide" formula_64 (the factor 5 in the numerator of the fraction has to be viewed as a "shift" of the -adic valuation, and thus it is not involved in the "division"). Multiplying Bézout's identity by formula_65 gives


title: P-adic number

text: The "integer part" formula_67 is not in the right interval. So, one has to use Euclidean division by formula_68 for getting formula_69 giving


title: P-adic number

text: As the "remainder" formula_74 has already been found, the process can be continued easily, giving coefficients formula_75 for odd powers of five, and formula_76 for even powers.


title: P-adic number

text: where every nonzero formula_4 is a rational number formula_81 such that none of formula_82 and formula_83 is divisible by .


title: P-adic number

text: Every rational number may be viewed as a -adic series with a single term, consisting of its factorization of the form formula_84 with and both coprime with .


title: P-adic number

text: A -adic series is "normalized" if each formula_4 is an integer in the interval formula_86 So, the -adic expansion of a rational number is a normalized -adic series.


title: P-adic number

text: The -adic valuation, or -adic order of a nonzero -adic series is the lowest integer such that formula_87 The order of the zero series is infinity formula_88


title: P-adic number

text: Two -adic series are "equivalent" if they have the same order , and if for every integer the difference between their partial sums


title: P-adic number

text: has an order greater than (that is, is a rational number of the form formula_90 with formula_91 and and both coprime with ).


title: P-adic number

text: For every -adic series formula_92, there is a unique normalized series formula_93 such that formula_92 and formula_93 are equivalent. formula_93 is the "normalization" of formula_97 The proof is similar to the existence proof of the -adic expansion of a rational number. In particular, every rational number can be considered as a -adic series with a single nonzero term, and the normalization of this series is exactly the rational representation of the rational number.


title: P-adic number

text: In other words, the equivalence of -adic series is an equivalence relation, and each equivalence class contains exactly one normalized -adic series.


title: P-adic number

text: The usual operations of series (addition, subtraction, multiplication, division) map -adic series to -adic series, and are compatible with equivalence of -adic series. That is, denoting the equivalence with , if , and are nonzero -adic series such that formula_98 one has


title: P-adic number

text: It is possible to use a positional notation similar to that which is used to represent numbers in base .


title: P-adic number

text: Let formula_100 be a normalized -adic series, i.e. each formula_4 is an integer in the interval formula_86 One can suppose that formula_103 by setting formula_104 for formula_105 (if formula_106), and adding the resulting zero terms to the series.


title: P-adic number

text: If formula_107 the positional notation consists of writing the formula_4 consecutively, ordered by decreasing values of , often with appearing on the right as an index:


title: P-adic number

text: When formula_112 a separating dot is added before the digits with negative index, and, if the index is present, it appears just after the separating dot. For example,


title: P-adic number

text: If a -adic representation is finite on the left (that is, formula_104 for large values of ), then it has the value of a nonnegative rational number of the form formula_116 with formula_117 integers. These rational numbers are exactly the nonnegative rational numbers that have a finite representation in base . For these rational numbers, the two representations are the same.


title: P-adic number

text: There are several equivalent definitions of -adic numbers. The one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections. Other equivalent definitions use completion of a discrete valuation ring (see ), completion of a metric space (see ), or inverse limits (see ).


title: P-adic number

text: A -adic number can be defined as a "normalized -adic series". Since there are other equivalent definitions that are commonly used, one says often that a normalized -adic series "represents" a -adic number, instead of saying that it "is" a -adic number.


title: P-adic number

text: One can say also that any -adic series represents a -adic number, since every -adic series is equivalent to a unique normalized -adic series. This is useful for defining operations (addition, subtraction, multiplication, division) of -adic numbers: the result of such an operation is obtained by normalizing the result of the corresponding operation on series. This well defines operations on -adic numbers, since the series operations are compatible with equivalence of -adic series.


title: P-adic number

text: With these operations, -adic numbers form a field called the field of -adic numbers and denoted formula_118 or formula_119 There is a unique field homomorphism from the rational numbers into the -adic numbers, which maps a rational number to its -adic expansion. The image of this homomorphism is commonly identified with the field of rational numbers. This allows considering the -adic numbers as an extension field of the rational numbers, and the rational numbers as a subfield of the -adic numbers.


title: P-adic number

text: The "valuation" of a nonzero -adic number , commonly denoted formula_120 is the exponent of in the first nonzero term of every -adic series that represents . By convention, formula_121 that is, the valuation of zero is formula_88 This valuation is a discrete valuation. The restriction of this valuation to the rational numbers is the -adic valuation of formula_123 that is, the exponent in the factorization of a rational number as formula_124 with both and coprime with .


title: P-adic number

text: Every integer is a -adic integer (including zero, since formula_127). The rational numbers of the form formula_128 with coprime with and formula_129 are also -adic integers (for the reason that has an inverse mod for every ).


title: P-adic number

text: The -adic integers form a commutative ring, denoted formula_130 or formula_131, that has the following properties.


title: P-adic number

text: The last property provides a definition of the -adic numbers that is equivalent to the above one: the field of the -adic numbers is the field of fractions of the completion of the localization of the integers at the prime ideal generated by .


title: P-adic number

text: The -adic valuation allows defining an absolute value on -adic numbers: the -adic absolute value of a nonzero -adic number is


title: P-adic number

text: where formula_137 is the -adic valuation of . The -adic absolute value of formula_138 is formula_139 This is an absolute value that satisfies the strong triangle inequality since, for every and one has


title: P-adic number

text: This makes the -adic numbers a metric space, and even an ultrametric space, with the -adic distance defined by


title: P-adic number

text: As a metric space, the -adic numbers form the completion of the rational numbers equipped with the -adic absolute value. This provides another way for defining the -adic numbers. However, the general construction of a completion can be simplified in this case, because the metric is defined by a discrete valuation (in short, one can extract from every Cauchy sequence a subsequence such that the differences between two consecutive terms have strictly decreasing absolute values; such a subsequence is the sequence of the partial sums of a -adic series, and thus a unique normalized -adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized -adic series instead of equivalence classes of Cauchy sequences).


title: P-adic number

text: As the metric is defined from a discrete valuation, every open ball is also closed. More precisely, the open ball formula_147 equals the closed ball formula_148 where is the least integer such that formula_149 Similarly, formula_150 where is the greatest integer such that formula_151


title: P-adic number

text: This implies that the -adic numbers form a locally compact space, and the -adic integers—that is, the ball formula_152—form a compact space.


title: P-adic number

text: The quotient ring formula_153 may be identified with the ring formula_154 of the integers modulo formula_155 This can be shown by remarking that every -adic integer, represented by its normalized -adic series, is congruent modulo formula_156 with its partial sum formula_157 whose value is an integer in the interval formula_158 A straightforward verification shows that this defines a ring isomorphism from formula_153 to formula_160


title: P-adic number

text: The inverse limit of the rings formula_153 is defined as the ring formed by the sequences formula_162 such that formula_163 and formula_164 for every .


title: P-adic number

text: The mapping that maps a normalized -adic series to the sequence of its partial sums is a ring isomorphism from formula_130 to the inverse limit of the formula_166 This provides another way for defining -adic integers (up to an isomorphism).


title: P-adic number

text: This definition of -adic integers is specially useful for practical computations, as allowing building -adic integers by successive approximations.


title: P-adic number

text: For example, for computing the -adic (multiplicative) inverse of an integer, one can use Newton's method, starting from the inverse modulo ; then, each Newton step computes the inverse modulo formula_167 from the inverse modulo formula_168


title: P-adic number

text: The same method can be used for computing the -adic square root of an integer that is a quadratic residue modulo . This seems to be the fastest known method for testing whether a large integer is a square: it suffices to test whether the given integer is the square of the value found in formula_153. Applying Newton's method to find the square root requires formula_170 to be larger than twice the given integer, which is quickly satisfied.


title: P-adic number

text: Hensel lifting is a similar method that allows to "lift" the factorization modulo of a polynomial with integer coefficients to a factorization modulo formula_170 for large values of . This is commonly used by polynomial factorization algorithms.


title: P-adic number

text: There are several different conventions for writing -adic expansions. So far this article has used a notation for -adic expansions in which powers of  increase from right to left. With this right-to-left notation the 3-adic expansion of , for example, is written as


title: P-adic number

text: When performing arithmetic in this notation, digits are carried to the left. It is also possible to write -adic expansions so that the powers of increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of is


title: P-adic number

text: -adic expansions may be written with other sets of digits instead of {0, 1, ..., }. For example, the 3-adic expansion of / can be written using balanced ternary digits {1,0,1} as


title: P-adic number

text: In fact any set of integers which are in distinct residue classes modulo may be used as -adic digits. In number theory, Teichmüller representatives are sometimes used as digits.


title: P-adic number

text: Both formula_130 and formula_118 are uncountable and have the cardinality of the continuum. For formula_177 this results from the -adic representation, which defines a bijection of formula_130 on the power set formula_179 For formula_118 this results from its expression as a countably infinite union of copies of formula_130: