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
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 usually 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, have no roots such as X2+1 over R, the real numbers. By constructing the splitting field for such a polynomial one can find the roots of the polynomial in the new field.
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:
- Factorize p(X) over Ki into irreducible factors .
- Choose any nonlinear irreducible factor f(X) = fi(X).
- Construct the field extension Ki+1 of Ki as the quotient ring Ki+1 = Ki[X]/(f(X)) where (f(X)) denotes the ideal in Ki[X] generated by f(X)
- Repeat the process for Ki+1 until p(X) completely factors.
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 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 ≤ j ≤ n−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 m ≥ n 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, x5 ≡ x, etc.; and so, for example p + qx + rx2 + sx3 ≡ p + qx + r⋅(−1) + s⋅(−x) = (p − r) + (q − s)⋅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, x5 ≡ x, 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 + bx → a + ib is a homomorphism with respect to addition and multiplication. It is also obvious that the map a + bx → a + ib is both injective and surjective; meaning that a + bx → a + ib is a bijective homomorphism, i.e. an isomorphism. It follows that, as claimed: R[x] / (x2 + 1) ≅ C.
Let K be the rational number field Q and
- p(X) = X3 − 2.
Each root of p equals 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
- The splitting field of x2 + 1 over F7 is F49; the polynomial has no roots in F7, i.e., −1 is not a square there, because 7 is not equivalent to 1 (mod 4).
- The splitting field of x2 − 1 over F7 is F7 since x2 − 1 = (x + 1)(x − 1) already factors into linear factors.
- We calculate the splitting field of f(x) = x3 + x + 1 over F2. It is easy to verify that f(x) has no roots in F2, hence f(x) is irreducible in F2[x]. Put r = x + (f(x)) in F2[x]/(f(x)) so F2(r) is a field and x3 + x + 1 = (x + r)(x2 + ax + b) in F2(r)[x]. Note that we can write + for − since the characteristic is two. Comparison of coefficients shows that a = r and b = 1 + r2. The elements of F2(r) can be listed as c + dr + er2, where c, d, e are in F2. There are eight elements: 0, 1, r, 1 + r, r2, 1 + r2, r + r2 and 1 + r + r2. Substituting these in x2 + rx + 1 + r2 we reach (r2)2 + r(r2) + 1 + r2 = r4 + r3 + 1 + r2 = 0, therefore x3 + x + 1 = (x + r)(x + r2)(x + (r + r2)) for r in F2[x]/(f(x)); E = F2(r) is a splitting field of x3 + x + 1 over F2.
- 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.
- Dummit, David S., and Foote, Richard M. (1999). Abstract Algebra (2nd ed.). New York: John Wiley & Sons, Inc. ISBN 0-471-36857-1.