A decisive Theorem (Un th\'eor\`eme d\'ecisif)

TL;DR

This paper provides an elementary proof that finite unramified algebras over discrete fields are étale, finite-dimensional, and tracically étale, clarifying foundational aspects of algebraic structures in field theory.

Contribution

It offers a simplified, elementary proof of a fundamental theorem relating unramified algebras and étale properties over discrete fields.

Findings

Finite unramified algebras over discrete fields are étale.

Such algebras are finite-dimensional vector spaces.

They are tracically étale.

Abstract

We give an elementary proof of the theorem which states that a finite unramified algebra over a discrete field is tracically \'etale. -- Nous donnons une d\'emonstration \'el\'ementaire du th\'eor\`eme selon lequel toute alg\`ebre nette sur un cors discret est \'etale, de dimension finie comme espace vectoriel et traciquement \'etale.