On three types of Galois extensions which are Azumaya

Abstract

In this paper, we shall study three types of Galois extensions [Formula: see text] which are Azumaya algebras over its center [Formula: see text]. (1) [Formula: see text], the class of Azumaya–Galois extensions; (2) [Formula: see text], the class of Galois extensions [Formula: see text] of [Formula: see text] with Galois group [Formula: see text] such that [Formula: see text] is a separable [Formula: see text]-algebra; and (3) [Formula: see text], the class of Galois extensions which are Azumaya algebras over its center [Formula: see text]. Clearly, [Formula: see text]. Examples are given to show the proper inclusion relationship [Formula: see text], and equivalent conditions are given under which a [Formula: see text] is in [Formula: see text] and a [Formula: see text] is in [Formula: see text].