D. Splitting Fields.
If F, B and E are three fields such that F C B C E, then we shall refer to B as an intermediate field.
If E is an extension of a field F in which a polynomial p(x) in F can be factored into linear factors, and if p(x) can not be so factored in any intermediate field, then we call E a splitting field for p(x). Thus, if E is a splitting field of p(x), the roots of p(x) generate E.
A splitting field is of finite degree since it is constructed by a finite number of adjunctions of algebraic elements, each defining an extension field of finite degree. Because of the corollary on page 22, the total degree is finite.
THEOREM 9. If p(x) is a polynomial in a field F, there exists a splitting field E of p(x).
We fantnr n (tc I in F into frre.Hiirfhle. fnrtnrs [f each of these is of the first degree then F itself is the required splitting field. Suppose then that f1(x) is of degree higher than the first. ByTheorem 7 there is an extension Fj of F in which f j( x ) has a root. Factor each of the factors f ( x), . . . , f ( x ) into irreducible factors in Fj and proceed as before. We finally arrive at a field in which p (x) can be split into linear factors. The field generated out of F by the roots of p(x) is the required splitting field.
The following theorem asserts that up to isomorphisms, the splitting field of a polynomial is unique.
THEOREM 10. Let Q be an isomorphism mapping the field F on the field F' , Let p (x ) be a polynomial in F and p 1 (x ) the polynomial in F 1 with coefficients corresponding to those of p (x ) under o. Finally, let E be a splitting field of p(x) and E' a splitting field of p' (x). Under these conditions the isomorphism g can be extended to an isomorphism between E and E 1 .
If f(x) is an irreducible factor of p(x) in F, then E contains a root of f( x ). For be the splitting of
p(x) in E. Then
f(x) as a polynomial in E and construct the extension field
in which f (a) Then
and a—a, being elements of the field B can have a product equal to 0 only if for one of the factors, say the first, we have a~a^ = 0. Thus, a = a^ , and a, is aroot of f(x).
Now in case all roots of p(x) are in F, then E = F and p(x) can be split in F. This factored form has an image in F' which is a splitting, of p1 (x), since the isomorphism g preserves all operations of addition and multiplication in the process of multiplying out the
factors of p(x) and collecting to get the original form. Since p ' (x) can be split in F' , we must have F ' = E . In this case, a itself is the required extension and the theorem is proved if all roots of p(x)
are in F-the polynomials corrrespondng to
We proceed by complete induction. Let us suppose the theorem proved for all cases in which the number of roots of p(x) outside of F is less than n > 1, and suppose that p (x ) is a polynomial having n roots outside of F. We factor p (x ) into irreducible factors in F; p(x) = f,(x) f2(x) • • f,(x)- Not all of these factors can be of degree 1, since otherwise p (x ) would split in F, contrary to assumption. Hence, we may suppose the degree of
be the factorization of p'(x) into is irreducible in F 1 , for a factorization of f J (x) in F 1 would induce 1) under a"1 a factorization of f,(x), which was however taken to be irreducible.
By Theorem 8, the isomorphism a can be extended to an isomorphism crj, between the fields F(a) and F 1 (a1).
Since F C F(a), p(x) is a polynomial in F(a) and E is a splitting field for p(x) in F(a). Similarly for p ' (x). There are now less than n roots of p (x ) outside the new ground field F (a). Hence by our inductive assumption a x can be extended from an isomorphism between F(a) and F 1 (a 1 ) to an isomorphism a2 between E and E 1 . Since tTj is an extension of <y, and a2 an extension of o we conclude (7j is an extension of (j and the theorem follows.
Corollary. If p(x) is a polynomial jn a field F, then any two Splitting fields for p (x ) are isomorphic.
This follows from Theorem 10 ^ we take F = F 1 and a to be the identity mapping, i.e., a(x) = x.
As a consequence of this corollary we see that we are justified in using the expression "the splitting field of p(x)" since any two differ only by an isomorphism. Thus, if p (x ) has repeated roots in one splitting field, so also in any other splitting field it will have repeated roots. The statement "p(x) has repeated roots" will be significant without reference to a particular splitting field.