Frattini Subgroup of the Unit Group of Central Simple Algebras

Abstract

Given an F-central simple algebra A = Mn(D), denote by A′ the derived group of its unit group A*. Here, the Frattini subgroup Φ(A*) of A* for various fields F is investigated. For global fields, it is proved that when F is a real global field, then Φ(A*) = Φ(F*)Z(A′), otherwise Φ(A*) = ⋂p∤ deg (A) F*p. Furthermore, it is also shown that Φ(A*) = k* whenever F is either a field of rational functions over a divisible field k or a finitely generated extension of an algebraically closed field k.

About Some Split Central Simple Algebras

“…Quaternion algebras and symbol algebras are central simple algebras. Quaternion algebras and symbol algebras have been studied from several points of view: from the theory of associative algebras ( [20], [12], [6], [7], [9], [16], [23]), from number theory ( [18], [12], [21], [15]), analysis and mecanics ( [14]). In this paper, we will determine certain split quaternion algebras and split symbol algebras, using some results of number theory (ramification theory in algebraic number fields, class field theory).…”

About some split central simple algebras

“…Quaternion algebras and symbol algebras have been studied from several points of view: from the theory of associative algebras ( [20], [12], [6], [7], [9], [16], [23]), from number theory ( [18], [12], [21], [15]), analysis and mecanics ( [14]). In this paper, we will determine certain split quaternion algebras and split symbol algebras, using some results of number theory (ramification theory in algebraic number fields, class field theory).…”