Algebraic number field

In mathematics, an algebraic number field (or simply number field) is an extension field formula_1 of the field of rational numbers such that the field extension formula_2 has finite degree (and hence is an algebraic field extension).
Thus formula_1 is a field that contains formula_4 and has finite dimension when considered as a vector space over 
The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
Definition.
Prerequisites.
The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplication.
Another notion needed to define algebraic number fields is vector spaces. To the extent needed here, vector spaces can be thought of as consisting of sequences (or tuples)
whose entries are elements of a fixed field, such as the field Any two such sequences can be added by adding the entries one per one. Furthermore, any sequence can be multiplied by a single element "c" of the fixed field. These two operations known as vector addition and scalar multiplication satisfy a number of properties that serve to define vector spaces abstractly. Vector spaces are allowed to be "infinite-dimensional", that is to say that the sequences constituting the vector spaces are of infinite length. If, however, the vector space consists of "finite" sequences
the vector space is said to be of finite dimension, "n".
Definition.
An algebraic number field (or simply number field) is a finite-degree field extension of the field of rational numbers. Here degree means the dimension of the field as a vector space over 
Algebraicity and ring of integers.
Generally, in abstract algebra, a field extension formula_27 is algebraic if every element formula_28 of the bigger field formula_1 is the zero of a polynomial with coefficients formula_30 in 
Every field extension of finite degree is algebraic. (Proof: for formula_32 in simply consider formula_33 – we get a linear dependence, i.e. a polynomial that formula_32 is a root of.) In particular this applies to algebraic number fields, so any element formula_28 of an algebraic number field formula_1 can be written as a zero of a polynomial with rational coefficients. Therefore, elements of formula_1 are also referred to as "algebraic numbers". Given a polynomial formula_38 such that formula_39, it can be arranged such that the leading coefficient formula_40 is one, by dividing all coefficients by it, if necessary. A polynomial with this property is known as a monic polynomial. In general it will have rational coefficients. If, however, its coefficients are actually all integers, formula_28 is called an "algebraic integer". Any (usual) integer formula_42 is an algebraic integer, as it is the zero of the linear monic polynomial:
It can be shown that any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer. It follows that the algebraic integers in formula_1 form a ring denoted formula_45 called the ring of integers of It is a subring of (that is, a ring contained in) A field contains no zero divisors and this property is inherited by any subring, so the ring of integers of formula_1 is an integral domain. The field formula_1 is the field of fractions of the integral domain This way one can get back and forth between the algebraic number field formula_1 and its ring of integers Rings of algebraic integers have three distinctive properties: firstly, formula_45 is an integral domain that is integrally closed in its field of fractions Secondly, formula_45 is a Noetherian ring. Finally, every nonzero prime ideal of formula_45 is maximal or, equivalently, the Krull dimension of this ring is one. An abstract commutative ring with these three properties is called a "Dedekind ring" (or "Dedekind domain"), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.
Unique factorization.
For general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product of prime ideals. For example, the ideal formula_52 in the ring formula_53 of quadratic integers factors into prime ideals as
However, unlike formula_55 as the ring of integers of the ring of integers of a proper extension of formula_56 need not admit unique factorization of numbers into a product of prime numbers or, more precisely, prime elements. This happens already for quadratic integers, for example in the uniqueness of the factorization fails:
Using the norm it can be shown that these two factorization are actually inequivalent in the sense that the factors do not just differ by a unit in Euclidean domains are unique factorization domains; for example the ring of Gaussian integers, and the ring of Eisenstein integers, where formula_58 is a cube root of unity (unequal to 1), have this property.
ζ-functions, "L"-functions and class number formula.
The failure of unique factorization is measured by the class number, commonly denoted "h", the cardinality of the so-called ideal class group. This group is always finite. The ring of integers formula_45 possesses unique factorization if and only if it is a principal ring or, equivalently, if formula_1 has class number 1. Given a number field, the class number is often difficult to compute. The class number problem, going back to Gauss, is concerned with the existence of imaginary quadratic number fields (i.e., formula_61) with prescribed class number. The class number formula relates "h" to other fundamental invariants of It involves the Dedekind zeta function ζformula_1(s), a function in a complex variable "s", defined by
(The product is over all prime ideals of formula_64 denotes the norm of the prime ideal or, equivalently, the (finite) number of elements in the residue field The infinite product converges only for Re("s") > 1, in general analytic continuation and the functional equation for the zeta-function are needed to define the function for all "s").
The Dedekind zeta-function generalizes the Riemann zeta-function in that ζformula_4("s") = ζ("s").
The class number formula states that ζformula_1("s") has a simple pole at "s" = 1 and at this point the residue is given by
Here "r"1 and "r"2 classically denote the number of real embeddings and pairs of complex embeddings of respectively. Moreover, Reg is the regulator of "w" the number of roots of unity in formula_1 and "D" is the discriminant of 
Dirichlet L-functions formula_69 are a more refined variant of formula_70. Both types of functions encode the arithmetic behavior of formula_4 and formula_1, respectively. For example, Dirichlet's theorem asserts that in any arithmetic progression
with coprime formula_74 and formula_75, there are infinitely many prime numbers. This theorem is implied by the fact that the Dirichlet formula_76-function is nonzero at formula_77. Using much more advanced techniques including algebraic K-theory and Tamagawa measures, modern number theory deals with a description, if largely conjectural (see Tamagawa number conjecture), of values of more general L-functions.
Bases for number fields.
Integral basis.
An "integral basis" for a number field formula_1 of degree formula_17 is a set
of "n" algebraic integers in formula_1 such that every element of the ring of integers formula_45 of formula_1 can be written uniquely as a Z-linear combination of elements of "B"; that is, for any "x" in formula_45 we have
where the "mi" are (ordinary) integers. It is then also the case that any element of formula_1 can be written uniquely as
where now the "mi" are rational numbers. The algebraic integers of formula_1 are then precisely those elements of formula_1 where the "mi" are all integers.
Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis, and it is now standard for computer algebra systems to have built-in programs to do this.
Power basis.
Let formula_1 be a number field of degree Among all possible bases of formula_1 (seen as a formula_4-vector space), there are particular ones known as power bases, that are bases of the form
for some element By the primitive element theorem, there exists such an formula_32, called a primitive element. If formula_32 can be chosen in formula_45 and such that formula_94 is a basis of formula_45 as a free Z-module, then formula_94 is called a power integral basis, and the field formula_1 is called a monogenic field. An example of a number field that is not monogenic was first given by Dedekind. His example is the field obtained by adjoining a root of the polynomial
formula_98
Regular representation, trace and discriminant.
Recall that any field extension formula_99 has a unique formula_4-vector space structure. Using the multiplication in formula_1, an element formula_32 of the field formula_1 over the base field formula_4 may be represented by formula_105 matrices
by requiring
Here formula_108 is a fixed basis for formula_1, viewed as a formula_4-vector space. The rational numbers formula_111 are uniquely determined by formula_32 and the choice of a basis since any element of formula_1 can be uniquely represented as a linear combination of the basis elements. This way of associating a matrix to any element of the field formula_1 is called the "regular representation". The square matrix formula_115 represents the effect of multiplication by formula_32 in the given basis. It follows that if the element formula_117 of formula_1 is represented by a matrix formula_119, then the product formula_120 is represented by the matrix product formula_121. Invariants of matrices, such as the trace, determinant, and characteristic polynomial, depend solely on the field element formula_32 and not on the basis. In particular, the trace of the matrix formula_123 is called the "trace" of the field element formula_32 and denoted formula_125, and the determinant is called the "norm" of "x" and denoted formula_126.
Now this can be generalized slightly by instead considering a field extension formula_127 and giving an formula_76-basis for formula_1. Then, there is an associated matrix formula_130 which has trace formula_131 and norm formula_132 defined as the trace and determinant of the matrix formula_130.
Example.
Consider the field extension formula_134 where formula_135. Then, we have a formula_4-basis given byformula_137since any formula_138 can be expressed as some formula_4-linear combinationformula_140Then, we can take some formula_141 where formula_142 and compute formula_143. Writing this out givesformula_144We can find the matrix formula_123 by writing out the associated matrix equation giving formula_146showingformula_147We can then compute the trace and determinant with relative ease, giving the trace and norm.
Properties.
By definition, standard properties of traces and determinants of matrices carry over to Tr and N: Tr("x") is a linear function of "x", as expressed by , , and the norm is a multiplicative homogeneous function of degree "n": , . Here "λ" is a rational number, and "x", "y" are any two elements of 
The "trace form" derived is a bilinear form defined by means of the trace, asformula_148 by formula_149formula_131. The integral trace form, an integer-valued symmetric matrix is defined as formula_151, where "b"1, …, "b"n is an integral basis for The "discriminant" of formula_1 is defined as det("t"). It is an integer, and is an invariant property of the field formula_1, not depending on the choice of integral basis.
The matrix associated to an element "x" of formula_1 can also be used to give other, equivalent descriptions of algebraic integers. An element "x" of formula_1 is an algebraic integer if and only if the characteristic polynomial "p""A" of the matrix "A" associated to "x" is a monic polynomial with integer coefficients. Suppose that the matrix "A" that represents an element "x" has integer entries in some basis "e". By the Cayley–Hamilton theorem, "p""A"("A") = 0, and it follows that "p""A"("x") = 0, so that "x" is an algebraic integer. Conversely, if "x" is an element of formula_1 which is a root of a monic polynomial with integer coefficients then the same property holds for the corresponding matrix "A". In this case it can be proven that "A" is an integer matrix in a suitable basis of The property of being an algebraic integer is "defined" in a way that is independent of a choice of a basis in 
Example with integral basis.
Consider formula_157, where "x" satisfies . Then an integral basis is [1, "x", 1/2("x"2 + 1)], and the corresponding integral trace form is
The "3" in the upper left hand corner of this matrix is the trace of the matrix of the map defined by the first basis element (1) in the regular representation of formula_1 on This basis element induces the identity map on the 3-dimensional vector space, The trace of the matrix of the identity map on a 3-dimensional vector space is 3.
The determinant of this is , the field discriminant; in comparison the root discriminant, or discriminant of the polynomial, is .
Places.
Mathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number. This situation changed with the discovery of p-adic numbers by Hensel in 1897; and now it is standard to consider all of the various possible embeddings of a number field formula_1 into its various topological completions formula_161 at once.
A "place" of a number field formula_1 is an equivalence class of absolute values on formula_1pg 9. Essentially, an absolute value is a notion to measure the size of elements formula_32 of Two such absolute values are considered equivalent if they give rise to the same notion of smallness (or proximity). The equivalence relation between absolute values formula_165 is given by some formula_166 such thatformula_167meaning we take the value of the norm formula_168 to the formula_169-th power.
For the following non-trivial norms occur (Ostrowski's theorem): the (usual) absolute value, sometimes denoted formula_178 which gives rise to the complete topological field of the real numbers On the other hand, for any prime number formula_38, the "p"-adic absolute values is defined by
are used to construct the formula_38-adic numbers In contrast to the usual absolute value, the "p"-adic norm gets "smaller" when "q" is multiplied by "p", leading to quite different behavior of formula_181 vis-à-vis 
Note the general situation typically considered is taking a number field formula_1 and considering a prime ideal formula_183 for its associated ring of algebraic numbers Then, there will be a unique place formula_184 called a non-archimedian place. In addition, for every embedding formula_185 there will be a place called an archimedian place, denoted This statement is a theorem also called Ostrowski's theorem.
Examples.
The field formula_186 for formula_187 where formula_188 is a fixed 6th root of unity, provides a rich example for constructing explicit real and complex archimedian embeddings, and non-archimedian embeddings as wellpg 15-16.
Archimedean places.
Here we use the standard notation formula_189 and formula_190 for the number of real and complex embeddings used, respectively (see below).
Calculating the archimedean places of a number field formula_1 is done as follows: let formula_32 be a primitive element of formula_1, with minimal polynomial formula_28 (over formula_4). Over formula_23, formula_28 will generally no longer be irreducible, but its irreducible (real) factors are either of degree one or two. Since there are no repeated roots, there are no repeated factors. The roots formula_198 of factors of degree one are necessarily real, and replacing formula_32 by formula_198 gives an embedding of formula_1 into formula_23; the number of such embeddings is equal to the number of real roots of Restricting the standard absolute value on formula_23 to formula_1 gives an archimedean absolute value on formula_1; such an absolute value is also referred to as a real place of On the other hand, the roots of factors of degree two are pairs of conjugate complex numbers, which allows for two conjugate embeddings into Either one of this pair of embeddings can be used to define an absolute value on formula_1, which is the same for both embeddings since they are conjugate. This absolute value is called a complex place of 
If all roots of formula_28 above are real (respectively, complex) or, equivalently, any possible embedding formula_208 is actually forced to be inside formula_23 (resp. formula_1 is called totally real (resp. totally complex).
Non-Archimedean or ultrametric places.
To find the non-Archimedean places, let again formula_28 and formula_32 be as above. In formula_28 splits in factors of various degrees, none of which are repeated, and the degrees of which add up to the degree of For each of these formula_38-adically irreducible factors we may suppose that formula_32 satisfies formula_216 and obtain an embedding of formula_1 into an algebraic extension of finite degree over Such a local field behaves in many ways like a number field, and the formula_38-adic numbers may similarly play the role of the rationals; in particular, we can define the norm and trace in exactly the same way, now giving functions mapping to By using this formula_38-adic norm map formula_220 for the place formula_216, we may define an absolute value corresponding to a given formula_38-adically irreducible factor formula_216 of degree formula_75 byformula_225Such an absolute value is called an ultrametric, non-Archimedean or formula_38-adic place of 
For any ultrametric place "v" we have that |"x"|"v" ≤ 1 for any "x" in since the minimal polynomial for "x" has integer factors, and hence its "p"-adic factorization has factors in Z"p". Consequently, the norm term (constant term) for each factor is a "p"-adic integer, and one of these is the integer used for defining the absolute value for "v".
Prime ideals in "OK".
For an ultrametric place "v", the subset of formula_45 defined by |"x"|"v" < 1 is an ideal formula_228 of This relies on the ultrametricity of "v": given "x" and "y" in then
Actually, formula_228 is even a prime ideal.
Conversely, given a prime ideal formula_228 of a discrete valuation can be defined by setting formula_231 where "n" is the biggest integer such that the "n"-fold power of the ideal. This valuation can be turned into an ultrametric place. Under this correspondence, (equivalence classes) of ultrametric places of formula_1 correspond to prime ideals of For this gives back Ostrowski's theorem: any prime ideal in Z (which is necessarily by a single prime number) corresponds to a non-Archimedean place and vice versa. However, for more general number fields, the situation becomes more involved, as will be explained below.
Yet another, equivalent way of describing ultrametric places is by means of localizations of Given an ultrametric place formula_233 on a number field the corresponding localization is the subring formula_234 of formula_1 of all elements formula_32 such that | "x" |"v" ≤ 1. By the ultrametric property formula_234 is a ring. Moreover, it contains For every element "x" of at least one of "x" or "x"−1 is contained in Actually, since "K"×/"T"× can be shown to be isomorphic to the integers, formula_234 is a discrete valuation ring, in particular a local ring. Actually, formula_234 is just the localization of formula_45 at the prime ideal so </math>.}} Conversely, formula_228 is the maximal ideal of 
Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations on a number field.
Lying over theorem and places.
One of the basic theorems in algebraic number theory are the going up and going down theorems, which describe the behavior of some prime ideal formula_183 when it is extended as an ideal in formula_243 for some field extension We say that an ideal formula_244 lies over formula_228 if Then, one incarnation of the theorem states a prime ideal in formula_246 lies over hence there is always a surjective mapformula_247induced from the inclusion Since there exists a correspondence between places and prime ideals, this means we can find places dividing a place which is induced from a field extension. That is, if formula_38 is a place of then there are places formula_233 of formula_76 which divide in the sense that their induced prime ideals divide the induced prime ideal of formula_38 in 
In fact, this observation is usefulpg 13 while looking at the base change of an algebraic field extension of formula_4 to one of its completions If we writeformula_253and write formula_254 for the induced element of we get a decomposition of Explicitly, this decomposition isformula_255furthermore, the induced polynomial formula_256 decomposes asformula_257because of Hensel's lemmapg 129-131, henceformula_258Moreover, there are embeddingsformula_259where formula_260 is a root of formula_261 giving formula_262, hence we could writeformula_263as subsets of formula_264 (which is the completion of the algebraic closure of 
Ramification.
Ramification, generally speaking, describes a geometric phenomenon that can occur with finite-to-one maps (that is, maps formula_265 such that the preimages of all points "y" in "Y" consist only of finitely many points): the cardinality of the fibers "f"−1("y") will generally have the same number of points, but it occurs that, in special points "y", this number drops. For example, the map
has "n" points in each fiber over "t", namely the "n" (complex) roots of "t", except in t = "0", where the fiber consists of only one element, "z" = 0. One says that the map is "ramified" in zero. This is an example of a branched covering of Riemann surfaces. This intuition also serves to define ramification in algebraic number theory. Given a (necessarily finite) extension of number fields formula_127, a prime ideal "p" of formula_243 generates the ideal "pO""K" of This ideal may or may not be a prime ideal, but, according to the Lasker–Noether theorem (see above), always is given by
with uniquely determined prime ideals "q""i" of formula_45 and numbers (called ramification indices) "e""i". Whenever one ramification index is bigger than one, the prime "p" is said to ramify in 
The connection between this definition and the geometric situation is delivered by the map of spectra of rings In fact, unramified morphisms of schemes in algebraic geometry are a direct generalization of unramified extensions of number fields.
Ramification is a purely local property, i.e., depends only on the completions around the primes "p" and "q""i". The inertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields.
An example.
The following example illustrates the notions introduced above. In order to compute the ramification index of where
at 23, it suffices to consider the field extension Up to 529 = 232 (i.e., modulo 529) "f" can be factored as
Substituting in the first factor "g" modulo 529 yields "y" + 191, so the valuation | "y" |"g" for "y" given by "g" is | −191 |23 = 1. On the other hand, the same substitution in "h" yields Since 161 = 7 × 23,
Since possible values for the absolute value of the place defined by the factor "h" are not confined to integer powers of 23, but instead are integer powers of the square root of 23, the ramification index of the field extension at 23 is two.
The valuations of any element of formula_1 can be computed in this way using resultants. If, for example "y" = "x"2 − "x" − 1, using the resultant to eliminate "x" between this relationship and "f" = "x"3 − "x" − 1 = 0 gives . If instead we eliminate with respect to the factors "g" and "h" of "f", we obtain the corresponding factors for the polynomial for "y", and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of "y" for "g" and "h" (which are both 1 in this instance.)
Dedekind discriminant theorem.
Much of the significance of the discriminant lies in the fact that ramified ultrametric places are all places obtained from factorizations in formula_181 where "p" divides the discriminant. This is even true of the polynomial discriminant; however the converse is also true, that if a prime "p" divides the discriminant, then there is a "p"-place which ramifies. For this converse the field discriminant is needed. This is the Dedekind discriminant theorem. In the example above, the discriminant of the number field formula_274 with "x"3 − "x" − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place which does. The other ramified place comes from the absolute value on the complex embedding of formula_1.
Galois groups and Galois cohomology.
Generally in abstract algebra, field extensions "K" / "L" can be studied by examining the Galois group Gal("K" / "L"), consisting of field automorphisms of formula_1 leaving formula_76 elementwise fixed. As an example, the Galois group formula_278 of the cyclotomic field extension of degree "n" (see above) is given by (Z/"nZ)×, the group of invertible elements in Z/"nZ. This is the first stepstone into Iwasawa theory.
In order to include all possible extensions having certain properties, the Galois group concept is commonly applied to the (infinite) field extension / "K" of the algebraic closure, leading to the absolute Galois group "G" := Gal( / "K") or just Gal("K"), and to the extension formula_2. The fundamental theorem of Galois theory links fields in between formula_1 and its algebraic closure and closed subgroups of Gal("K"). For example, the abelianization (the biggest abelian quotient) "G"ab of "G" corresponds to a field referred to as the maximal abelian extension "K"ab (called so since any further extension is not abelian, i.e., does not have an abelian Galois group). By the Kronecker–Weber theorem, the maximal abelian extension of formula_4 is the extension generated by all roots of unity. For more general number fields, class field theory, specifically the Artin reciprocity law gives an answer by describing "G"ab in terms of the idele class group. Also notable is the Hilbert class field, the maximal abelian unramified field extension of formula_1. It can be shown to be finite over formula_1, its Galois group over formula_1 is isomorphic to the class group of formula_1, in particular its degree equals the class number "h" of formula_1 (see above).
In certain situations, the Galois group acts on other mathematical objects, for example a group. Such a group is then also referred to as a Galois module. This enables the use of group cohomology for the Galois group Gal("K"), also known as Galois cohomology, which in the first place measures the failure of exactness of taking Gal("K")-invariants, but offers deeper insights (and questions) as well. For example, the Galois group "G" of a field extension "L" / "K" acts on "L"×, the nonzero elements of "L". This Galois module plays a significant role in many arithmetic dualities, such as Poitou-Tate duality. The Brauer group of originally conceived to classify division algebras over formula_1, can be recast as a cohomology group, namely H2(Gal ("K", ×).
Local-global principle.
Generally speaking, the term "local to global" refers to the idea that a global problem is first done at a local level, which tends to simplify the questions. Then, of course, the information gained in the local analysis has to be put together to get back to some global statement. For example, the notion of sheaves reifies that idea in topology and geometry.
Local and global fields.
Number fields share a great deal of similarity with another class of fields much used in algebraic geometry known as function fields of algebraic curves over finite fields. An example is "K""p"("T"). They are similar in many respects, for example in that number rings are one-dimensional regular rings, as are the coordinate rings (the quotient fields of which is the function field in question) of curves. Therefore, both types of field are called global fields. In accordance with the philosophy laid out above, they can be studied at a local level first, that is to say, by looking at the corresponding local fields. For number fields the local fields are the completions of formula_1 at all places, including the archimedean ones (see local analysis). For function fields, the local fields are completions of the local rings at all points of the curve for function fields.
Many results valid for function fields also hold, at least if reformulated properly, for number fields. However, the study of number fields often poses difficulties and phenomena not encountered in function fields. For example, in function fields, there is no dichotomy into non-archimedean and archimedean places. Nonetheless, function fields often serves as a source of intuition what should be expected in the number field case.
Hasse principle.
A prototypical question, posed at a global level, is whether some polynomial equation has a solution in If this is the case, this solution is also a solution in all completions. The local-global principle or Hasse principle asserts that for quadratic equations, the converse holds, as well. Thereby, checking whether such an equation has a solution can be done on all the completions of which is often easier, since analytic methods (classical analytic tools such as intermediate value theorem at the archimedean places and p-adic analysis at the nonarchimedean places) can be used. This implication does not hold, however, for more general types of equations. However, the idea of passing from local data to global ones proves fruitful in class field theory, for example, where local class field theory is used to obtain global insights mentioned above. This is also related to the fact that the Galois groups of the completions "K"v can be explicitly determined, whereas the Galois groups of global fields, even of formula_4 are far less understood.
Adeles and ideles.
In order to assemble local data pertaining to all local fields attached to the adele ring is set up. A multiplicative variant is referred to as ideles.