The geometry of the moduli Space of non-cyclic biquadratic field extensions

TL;DR

This paper explores the geometric structure of the moduli space of non-cyclic biquadratic field extensions, establishing their classification and providing polynomial families to describe quartic extensions.

Contribution

It introduces the existence of an elementary abelian closure for non-cyclic biquadratic extensions and describes the moduli space's geometry using group theory.

Findings

Existence of elementary abelian closure in non-cyclic biquadratic extensions

Classification of these extensions up to isomorphism via descent

Two families of parametric polynomials describing quartic extensions

Abstract

In this paper, we investigate the existence of an elementary abelian closure in characteristic not $2$ for biquadratic extensions. We discover that it exists for any non-cyclic extension. We make use of it to obtain a classification for this class of extensions up to isomorphism via descent. This permits us to describe the geometry of this moduli space in group theoretic terms. We also provide two families of polynomials with two parameters that can describe any quartic extensions.