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Abstract
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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.
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