The Structure of Galois Algebras

Abstract

Let B be a ring with 1 and G an automorphism group of B of order n for some integer n. It is shown that if B is a Galois algebra with Galois group G, then B is either a direct sum of central Galois algebras or a direct sum of central Galois algebras and a commutative Galois algebra. Moreover, when G is inner, B is either a direct sum of Azumaya projective group algebras or a direct sum of Azumaya projective group algebras and a commutative Galois algebra. Examples are given for these structures.

A Generalization of the Demeyer Theorem for Central Galois Algebras

“…Proof. By Theorem 3.8 in [7], there exist central idempotents fE j j j = 1; 2; :::; n for some integer ng such that B = BE 0¨(¨P n j=1 BE j ) where BE j is a central Galois algebra over CE j with Galois group H j contained in G for each j = 1; 2; :::; n and BE 0 is a commutative Galois algebra over RE 0 with Galois group Gj BE 0 $ = G. Since RE j is a semi-local ring, CE j is a semi-local ring; and so BE j is a projective group algebra for each j = 1; 2; :::; n by Theorem 3.7.…”

“…We note that the condition in Theorem 3.9, G(e i ) T = f1g, is important to have a nontrivial Galois algebra Be i over Re i . In case G(e i ) = f1g for some i, we shall employ the structure theorem as given in [7] for B to avoid this situation. Theorem 3.10.…”

On Hirata-Azumaya Galois extensions

“…In [3], [6], [9], the class of central Galois extensions B over C with Galois group G was studied. In [5], [8], the class of Hirata separable and Galois extensions B of B G with Galois group G was investigated.…”