Brauer-type results on semigroups over p-adic fields

In this paper we show that every central simple algebra A over Q p , generated by a multiplicative semigroup S ⊂ A with the property that the minimal polynomial of every element in S splits over Q p , is isomorphic to M n (Q p ). If, in addition, S C A* is a compact group, then it contains a commutative normal subgroup of finite index.