Minkowski's second theorem over a simple algebra (Q857838)

Minkowski's second theorem asserts that the product of the successive minima of a lattice in the Euclidean space \({\mathbb R}^n\) is bounded from above by the product of the covolume of the lattice and the Hermite constant. \textit{T. Watanabe} [J. Math. Soc Japan 55, 1061--1080 (2003; Zbl 1103.11033)] generalized this Hermite constant also to free modules over finite-dimensional simple algebras over global fields. The present paper proves an analogue of Minkowski's theorem in this situation. This theorem is applied to derive an analogue of Siegel's lemma on the integral solutions of a system of linear equations. The introduction contains a survey on related results.