Splitting field

In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial "splits" or decomposes into linear factors.
Definition.
A splitting field of a polynomial "p"("X") over a field "K" is a field extension "L" of "K" over which "p" factors into linear factors
where formula_2 and for each formula_3 we have formula_4 with "ai" not necessarily distinct and such that the roots "ai" generate "L" over "K". The extension "L" is then an extension of minimal degree over "K" in which "p" splits. It can be shown that such splitting fields exist and are unique up to isomorphism. The amount of freedom in that isomorphism is known as the Galois group of "p" (if we assume it is separable).
Properties.
An extension "L" which is a splitting field for a set of polynomials "p"("X") over "K" is called a normal extension of "K".
Given an algebraically closed field "A" containing "K", there is a unique splitting field "L" of "p" between "K" and "A", generated by the roots of "p". If "K" is a subfield of the complex numbers, the existence is immediate. On the other hand, the existence of algebraic closures in general is often proved by 'passing to the limit' from the splitting field result, which therefore requires an independent proof to avoid circular reasoning.
Given a separable extension "K"′ of "K", a Galois closure "L" of "K"′ is a type of splitting field, and also a Galois extension of "K" containing "K"′ that is minimal, in an obvious sense. Such a Galois closure should contain a splitting field for all the polynomials "p" over "K" that are minimal polynomials over "K" of elements "a" of "K"′.
Constructing splitting fields.
Motivation.
Finding roots of polynomials has been an important problem since the time of the ancient Greeks. Some polynomials, however, such as over , the real numbers, have no roots. By constructing the splitting field for such a polynomial one can find the roots of the polynomial in the new field.
The construction.
Let "F" be a field and "p"("X") be a polynomial in the polynomial ring "F"["X"] of degree "n". The general process for constructing "K", the splitting field of "p"("X") over "F", is to construct a chain of fields formula_5 such that "Ki" is an extension of "K""i"−1 containing a new root of "p"("X"). Since "p"("X") has at most "n" roots the construction will require at most "n" extensions. The steps for constructing "Ki" are given as follows:
The irreducible factor "fi"("X") used in the quotient construction may be chosen arbitrarily. Although different choices of factors may lead to different subfield sequences, the resulting splitting fields will be isomorphic.
Since "f"("X") is irreducible, ("f"("X")) is a maximal ideal of "K""i"["X"] and "K""i"["X"]/("f"("X")) is, in fact, a field. Moreover, if we let formula_7 be the natural projection of the ring onto its quotient then
so π("X") is a root of "f"("X") and of "p"("X").
The degree of a single extension formula_9 is equal to the degree of the irreducible factor "f"("X"). The degree of the extension ["K" : "F"] is given by formula_10 and is at most "n"!.
The field "K""i"["X"]/("f"("X")).
As mentioned above, the quotient ring "K""i"+1 = "K""i"["X"]/("f"("X")) is a field when "f"("X") is irreducible. Its elements are of the form
where the "cj" are in "Ki" and α = π("X"). (If one considers "K""i"+1 as a vector space over "Ki" then the powers α"j" for form a basis.)
The elements of "K""i"+1 can be considered as polynomials in α of degree less than "n". Addition in "K""i"+1 is given by the rules for polynomial addition and multiplication is given by polynomial multiplication modulo "f"("X"). That is, for "g"(α) and "h"(α) in "K""i"+1 the product "g"(α)"h"(α) = "r"(α) where "r"("X") is the remainder of "g"("X")"h"("X") divided by "f"("X") in "Ki"["X"].
The remainder "r"("X") can be computed through long division of polynomials, however there is also a straightforward reduction rule that can be used to compute "r"(α) = "g"(α)"h"(α) directly. First let
The polynomial is over a field so one can take "f"("X") to be monic without loss of generality. Now α is a root of "f"("X"), so
If the product "g"(α)"h"(α) has a term α"m" with it can be reduced as follows:
As an example of the reduction rule, take "Ki" = Q["X"], the ring of polynomials with rational coefficients, and take "f"("X") = "X"7 − 2. Let formula_15 and "h"(α) = α3 +1 be two elements of Q["X"]/("X"7 − 2). The reduction rule given by "f"("X") is α7 = 2 so
Examples.
The complex numbers.
Consider the polynomial ring R["x"], and the irreducible polynomial The quotient ring is given by the congruence As a result, the elements (or equivalence classes) of are of the form where "a" and "b" belong to R. To see this, note that since it follows that , , , etc.; and so, for example 
The addition and multiplication operations are given by firstly using ordinary polynomial addition and multiplication, but then reducing modulo , i.e. using the fact that , , , , etc. Thus:
If we identify with ("a","b") then we see that addition and multiplication are given by
We claim that, as a field, the quotient is isomorphic to the complex numbers, C. A general complex number is of the form , where "a" and "b" are real numbers and Addition and multiplication are given by
If we identify with ("a","b") then we see that addition and multiplication are given by
The previous calculations show that addition and multiplication behave the same way in and C. In fact, we see that the map between and C given by is a homomorphism with respect to addition "and" multiplication. It is also obvious that the map is both injective and surjective; meaning that is a bijective homomorphism, i.e. an isomorphism. It follows that, as claimed: 
In 1847, Cauchy used this approach to "define" the complex numbers.
Cubic example.
Let be the rational number field and . Each root of equals times a cube root of unity. Therefore, if we denote the cube roots of unity by
any field containing two distinct roots of will contain the quotient between two distinct cube roots of unity. Such a quotient is a primitive cube root of unity—either ω2 or formula_28. It follows that a splitting field of will contain ω2, as well as the real cube root of 2; conversely, any extension of containing these elements contains all the roots of . Thus
Note that applying the construction process outlined in the previous section to this example, one begins with formula_30 and constructs the field formula_31. This field is not the splitting field, but contains one (any) root. However, the polynomial formula_32 is not irreducible over formula_33 and in fact:
Note that formula_35 is not an indeterminate, and is in fact an element of formula_33. Now, continuing the process, we obtain formula_37 which is indeed the splitting field and is spanned by the formula_38-basis formula_39. Notice that if we compare this with formula_40 from above we can identify formula_41 and formula_42.