Abelian Galois groups

Abstract

The question of the existence of noninner, nonouter Abelian Galois groups of noncommutative rings seems not to have been considered previously. Amitsur [1 ] may have come closest when he constructed noninner, nonouter cyclic division ring extensions. Although these extensions are finite dimensional division ring extensions of Galois subrings corresponding to cyclic groups of automorphisms, nevertheless, as will be shown below, their Galois groups need not be Abelian. The question is made more critical by results obtained below which imply for any finite dimensional division algebra, and, furthermore, for any full matrix ring over it having center 7 GF(4), that any Abelian Galois group is inner or outer. Curiously enough Galois groups of the required kind do exist for some matrix rings over GF(4). Affirmative results are obtained also for certain infinite dimensional division algebras first constructed by Kothe [3 ], and their associated matrix rings. For these rings a general procedure for obtaining noninner, nonouter Abelian Galois groups is discussed.