FRATTINI SUBGROUP OF THE UNIT GROUP OF CENTRAL SIMPLE ALGEBRAS

Given an F-central simple algebra A = M-n(D), denote by A' the derived group of its unit group A*. Here, the Frattini subgroup Phi(A*) of A* for various fields F is investigated. For global fields, it is proved that when F is a real global field, then Phi(A*) = Phi(F*)Z(A'), otherwise Phi(A*) = boolean AND(p inverted iota deg(A)) F*(p). Furthermore, it is also shown that Phi(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.