The Fundamental Theorem of the Galois Theory for Quasi-Fields

It has been noted recently by Artin and by Baer that one obtains the fundamental theorem of the Galois theory most readily by starting with a finite group 65 of automorphisms in a field P and determining the structure of P over c1 the set of invariant elements. One proves that P is finite, separable and normal and (M is its Galois group over 4'. The theorem is completed by proving that any isomorphism between subfields over 4' in a finite, separable and normal extension P of 4' can be extended to an automorphism in P. It follows that the Galois group of P over 4' has 4' as its set of invariant elements. The correspondence between subfields and subgroups follows readily. The usual proofs of these theorems are obtained by strictly commutative methods (symmetric functions, unique factorization of polynomials). In the present paper we begin with an arbitrary quasi-field P and a finite group of outer automorphisms 5 acting in P and establish the correspondence between subgroups of (M and sub-quasi-fields Z between P and 4' the set of invariant elements. The methods are necessarily those of non-commutative algebra. The particular tool used is the theory of simple rings. We obtain some applications to division algebras. The first section is introductory containing results that are for the most part well known. In the last section we give a generalization of Hilbert's theorem on the elements of norm 1 in a cyclic field.