Galois Theory (Second Edition) by Emil Artin

By Emil Artin

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A ( xf ) by 3^. a(a) = 2 xtr(a) = a reG so that xa = a/a (a). A solution to the Noether equations defines a mapping C of G into E, namely, C ( a ) = xa. If F is the fixed field of G, and the elements xa lie in F, then C is a character of G. For C(or)•= x^ = x a - a ( x r ) = x^xf = C ( a ) - C ( r ) since a ( x f ) = x t if xr € F. Conversely, each character C of G in F provides a solution 59 to the Noether equations. Call C(a) = xa. Then, since xf £ F, we have a(x f ) = xr . Thus, x a - a ( x f ) = x a -x f = C ( a ) - C ( r ) = C(ar) = x^.

Conversely, C^1 . . C^1 defines a character. Since the order of C. , the character group X of G is isomorphic to G. If a 4 1, then in o = GJ l o2 2 . . , say i^ , is not divisible by m . 4 1. Let A denote the set of those non-zero elements a of E for which r a e F and let FI denote the non-zero elements of F. It is obvious that A is a multiplicative group and that FI is a subgroup of A. Let Ar denote the set of rth powers of elements in A and FJ the set of rth powers of elements of FX. The following theorem provides in most applications a convenient method for computing the group G.

Thus, B is a normal extension of F if and only if the number of automorphisms of B is (B/F). B is a normal extension of F if and only if each isomorphism of B into E is an automorphism of B. This follows from the fact that each of the above conditions are equivalent to the assertion that there are 49 the same number of isomorphisms and automorphisms. Since, for each a, B =

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