Algèbres simples centrales à involution de première espèce

Abstract

This article provides new and elementary proofs for some of the crucial theorems in the theory of central simple algebras with involution of the first kind. In the first place Albert's criterion for the existence of an involution of the first kind and Kneser's extension theorem for such involutions are presented in a unified way. These two results are retrieved as corollaries of a new theorem which gives a criterion to decide whether an antiautomorphism of a central simple algebra is an involution of the first kind. Two examples are given to indicate that the analogous approach cannot be applied to involutions of the second kind. Quaternion algebras give the easiest nontrivial examples of central simple algebras which carry an involution of the first kind. Albert has shown that any central simple algebra of dimension $16$ with involution of the first kind is a tensor product of two quaternion algebras. This theorem is presented here with a new proof essentially using basic linear algebra.