# Tores associ\'es \`a une alg\`ebre \'etale quartique

## TL;DR

This paper investigates homomorphisms between multiplicative groups of degree 4 and 6 etale algebras, relating norm groups and describing 2-torsion in the Brauer group, extending properties of biquadratic extensions.

## Contribution

It introduces new homomorphisms between multiplicative groups of degree 4 and 6 etale algebras, generalizing properties of biquadratic extensions to all such algebras.

## Key findings

- Relates norm groups via homomorphisms
- Describes 2-torsion in the Brauer group
- Extends properties of biquadratic extensions

## Abstract

Homomorphisms are defined between the multiplicative group of an etale algebra of dimension 4 and the multiplicative group of a canonically associated etale algebra of degree 6 over an arbitrary field. These homomorphisms are used to relate various norm groups and to describe the 2-torsion in the relative Brauer group of a separable extension of degree 4. Several remarkable properties of biquadratic extensions are thus extended to arbitrary etale algebras of dimension 4.