BRAUER-TYPE RESULTS ON SEMIGROUPS OVER p-ADIC FIELDS

In this paper we consider the following problem. Let k < K be fields. Suppose S ⊂ Mn(K) is an absolutely irreducible multiplicative semigroup with the property that the spectrum of every a ∈ S is contained in k. Does it follow that S is simultaneously realizable in Mn(k), that is, does there exist a p ∈ GLn(K) such that pSp−1 ⊂ Mn(k)? The overfield K plays no essential role in this situation. Consider the k-algebra A generated by S. It is easy to see that it is a central simple algebra over k (see [7]). If one can show that A is isomorphic to Mn(k), then this isomorphism can be extended to an inner isomorphism of Mn(K) by Skolem-Noether theorem. So we consider the more intrinsic version of the question above. Suppose A is a central simple algebra over k and S ⊂ A a multiplicative semigroup that generates A as a k-vector space and has the property that the minimal polynomial of every element in S splits over k. Does it follow A Mn(k)? If we exclude the case when the Brauer group of k is trivial, then the question has no apparent answer. If S is a finite subgroup of A∗ and char (k) = 0, then the answer is affirmative by Brauer’s theorem on splitting fields (see [3, Thm. 41.1]). The general semigroup case was considered for some particular fields. In [7] it is shown that the answer is affirmative in the special case k = R with no additional assumptions on S (see also [9] for some related results). In this paper we consider the case k = Qp for every rational prime number p. The proof proceeds in two steps. First we consider the special case when S is a compact subgroup of A∗. The crucial part in this case is the fact that the Lie algebra of S is commutative. This however is not true if k is a general p-adic field (see the example at the end of the paper). The second step is to reduce the problem from arbitrary semigroup with the desired property to