Constructive basic theory of central simple algebras

TL;DR

This paper develops a constructive approach to the theory of central simple algebras, replacing non-constructive results with dynamical versions to enable explicit proofs of fundamental theorems.

Contribution

It introduces a dynamical version of Wedderburn's Theorem, enabling constructive proofs of key results like Skolem-Noether and Becher's theorem.

Findings

Constructive proof of a dynamical version of Wedderburn's Theorem

Constructive proofs of Skolem-Noether and Becher's theorems

Enhanced understanding of central simple algebras through constructive methods

Abstract

We provide a constructive treatment of basic results in the theory of central simple algebras. One main issue is the fact that one starting result, Wedderburn's Theorem stating that a simple algebra is a matrix algebra over a skew field, is not constructively valid. We solve this problem by proving instead a dynamical version of this theorem. One can use this to give constructive proofs of basic results of the theory of central simple algebras, such as Skolem-Noether Theorem. We illustrate this development by giving an elementary constructive proof of a theorem of Becher (which is itself a consequence of a celebrated theorem of Merkurjev).