Brauer-type results on semigroups over \(p\)-adic fields.
Let \(k\) be a field, \(K\) an arbitrary extension of \(k\), and \(S\) an absolutely irreducible multiplicative subsemigroup of \(M_n(K)\) with the property that the spectrum of every element \(a\in S\) is contained in \(k\). The paper under review considers the problem of whether \(S\) is simultaneously realizable in \(k\), i.e. there exists an element \(\rho\in\text{GL}_n(K)\), such that \(\rho S\rho^{-1}\subset M_n(k)\). It shows that if \(k\) is the field of \(p\)-adic numbers, \(A\) is a central simple \(k\)-algebra, and \(S\) is a multiplicative subsemigroup of \(A\), such that \(A\) is spanned over \(k\) by \(S\), and the minimal polynomials of the elements of \(S\) split over \(k\), then \(A\) is isomorphic to the full matrix \(k\)-algebra \(M_n(k)\), for some positive integer \(n\). In addition, it proves that if \(S\) is a compact subgroup of the multiplicative group \(A^*\), then it possesses a commutative normal subgroup of finite index. In view of the classical Brauer-Hasse-Noether-Albert theorem, the former result is also true, if \(k\) is the field of rational numbers (a fact implicitly contained in [\textit{M. Radjabalipour} and \textit{H. Radjavi}, Commun. Algebra 27, No. 1, 301-319 (1999; Zbl 0920.20058)]). On the other hand, the author points out that his proof does not generalize to the case where \(k\) is a finite extension of the field of \(p\)-adic numbers.
scientific article; zbMATH DE number 2101048