Alg\`ebre lin\'eaire

TL;DR

This paper explores linear algebra over commutative rings, focusing on matrix calculus, determinants, and algebraic structures like skewfields, using classical and modern algebraic techniques.

Contribution

It presents a unified approach to matrix calculus, determinants, and algebraic structures over commutative rings, including a modified proof of factoriality and classical theorems without central simple algebras.

Findings

Factoriality of polynomial rings over integers established via modified Zermelo's proof.

Derivation of Wederburn and Erdös-Kaplanski theorems within this framework.

Application of Bourbaki's rational structures to algebraic classifications.

Abstract

After the language of module and theirs morphisms, this short course presents matricial calculus and determinants in a commutative ring as appliction of ``remarquable identities'' in the ring of polynomials with integer coefficients with variable coefficients and second member of the ``general Gauss method'' requiring the factoriality of such polynomial rings, here obtained by a modification of Zermolo's proof of the factoriality of the integers. As third part, using the, today almost forgotten, presentation in the first edition of N. Bourbaki's Algebra of rational structures for a subfield, one gets, without central simple algebras, the baba of skewfields, and the theorems of Wederburn and Erd\"os-Kaplanski (commutativity of finite fields and dimension of the dual).