Splitting field

This article is about splitting field of polynomial. For splitting field of CSA, see central simple algebra.

In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial splits or decomposes into linear factors.


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 for each we have

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).


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


Finding roots of polynomials has been an important problem since the time of the ancient Greeks. Some polynomials, however, such as x2 + 1 over R, 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 sequence of fields such that Ki is an extension of Ki−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 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 and hence Ki[X]/(f(X)) is, in fact, a field. Moreover, if we let 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 is equal to the degree of the irreducible factor f(X). The degree of the extension [K : F] is given by and is at most n!.

The field Ki[X]/(f(X))

As mentioned above, the quotient ring Ki+1 = Ki[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 Ki+1 as a vector space over Ki then the powers αj for 0 ≤ jn−1 form a basis.)

The elements of Ki+1 can be considered as polynomials in α of degree less than n. Addition in Ki+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 Ki+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 αm with mn 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) = X7 − 2. Let and h(α) = α3 +1 be two elements of Q[X]/(X7 − 2). The reduction rule given by f(X) is α7 = 2 so


The complex numbers

Consider the polynomial ring R[x], and the irreducible polynomial x2 + 1. The quotient ring R[x] / (x2 + 1) is given by the congruence x2 ≡ −1. As a result, the elements (or equivalence classes) of R[x] / (x2 + 1) are of the form a + bx where a and b belong to R. To see this, note that since x2 ≡ −1 it follows that x3 ≡ −x, x4 ≡ 1, x5x, etc.; and so, for example p + qx + rx2 + sx3p + qx + r⋅(−1) + s⋅(−x) = (pr) + (qs)⋅x.

The addition and multiplication operations are given by firstly using ordinary polynomial addition and multiplication, but then reducing modulo x2 + 1, i.e. using the fact that x2 ≡ −1, x3 ≡ −x, x4 ≡ 1, x5x, etc. Thus:

If we identify a + bx with (a,b) then we see that addition and multiplication are given by

We claim that, as a field, the quotient R[x] / (x2 + 1) is isomorphic to the complex numbers, C. A general complex number is of the form a + ib, where a and b are real numbers and i2 = −1. Addition and multiplication are given by

If we identify a + ib 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 R[x] / (x2 + 1) and C. In fact, we see that the map between R[x]/(x2 + 1) and C given by a + bxa + ib is a homomorphism with respect to addition and multiplication. It is also obvious that the map a + bxa + ib is both injective and surjective; meaning that a + bxa + ib is a bijective homomorphism, i.e. an isomorphism. It follows that, as claimed: R[x] / (x2 + 1) ≅ C.

In 1847, Cauchy used this approach to define the complex numbers.[1]

Cubic example

Let K be the rational number field Q and p(x) = x3 − 2. Each root of p equals 32 times a cube root of unity. Therefore, if we denote the cube roots of unity by


any field containing two distinct roots of p will contain the quotient between two distinct cube roots of unity. Such a quotient is a primitive cube root of unity—either ω2 or . It follows that a splitting field L of p will contain ω2, as well as the real cube root of 2; conversely, any extension of Q containing these elements contains all the roots of p. Thus

Note that applying the construction process outlined in the previous section to this example, one begins with and constructs the field . This field is not the splitting field, but contains one (any) root. However, the polynomial is not irreducible over and in fact, factorizes into . Note that is not an indeterminate, and is in fact an element of . Now, continuing the process, we obtain which is indeed the splitting field (and is spanned by the -basis ).

Other examples

See also


  1. Cauchy, Augustin-Louis (1847), "Mémoire sur la théorie des équivalences algébriques, substituée à la théorie des imaginaires", Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences (in French), 24: 1120–1130
  2. Instead of applying this characterization of odd prime moduli for which −1 is a square, one could just check that the set of squares in F7 is the set of classes of 0, 1, 4, and 2, which does not include the class of −1≡6.


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