N. Existence of a Normal Basis.
The following theorem is true for any field though we prove it only in the case that F contains an infinity of elements.
THEOREM 28. If E is a normal extension of F and o.t<j .,o
are the elements of its group G, there is an element $ in E such that
the n elements a. (6), a9(0),..., a ( 0) are linearly independent with
respect to F.
According to Theorem 27 there is an a such that
f(x) be the equation for a, put o,(a) = a
g j( x ) is a polynomial in E having as root for k ^ i and hence In the equation the left side is of degree at most n — 1. If (2) is true for n different values of x, the left side must be identically 0. Such n values are
Multiplying (2) by g^ x) and using (1) shows:
alta2, . . . ,a„ Since
and prove D(x) £ 0. If we square it by multiplying column by column and compute its value (mod f(x)) we get from (1), (2), (3) a determinant that has 1 in the diagonal and 0 elsewhere.
Consider anv linear relation where the x. are in F. Apply
D (x ) can have only a finite number of roots in F. Avoiding them we can find a value a for x such that D(a) 0. Now set 0 = g(a). Then the determinant
ing the automorphism <7. to it would lead to n homogeneous equations for the n unknowns x.. (5) shows that x. = 0 and our theorem is proved.
0. Theorem on Natural Irrationalities.
We next compute the determinant
Let F be a field, p(x) a polynomial in F whose irreducible factors are separable, and let E be a splitting field for p(x). Let B be an arbitrary extension of F, and let us denote by EB the splitting field of p(x) when p(x) is taken to lie in B. If alt . . . ra& are the roots of p(x) in EB, then F(a,, . . . ) is a subfield of EB which is readily seen to form a splitting field for p(x) in F. By Theorem 10, E and F(aJJ - - . , )are isomorphic. There is therefore no loss of generality if in the sequel we take and assume therefore that E is a subfield
Let us denote by the intersection of E and B. It is readily seen that E n B is a field and is intermediate to F and E.
THEOREM 29. If G is the group of automorphisms of E over F, and H the group of EB over B, then H is isomorphic to the subgroup of G having E H B as its fixed field.