hamidreza dorbidi

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∗) = Tpdeg(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.