The moduli space of $G$-algebras

TL;DR

This paper studies the moduli space of Galois algebras with a normal element, describing its structure as a quotient of a projective space and providing a height formula based on algebraic invariants.

Contribution

It characterizes the moduli space of Galois algebras with a normal element as an open subset of a quotient variety and derives a height formula using adelic metrics.

Findings

Moduli space is isomorphic to an open subset of a quotient of projective space.

Provides a formula for the height of pairs in the moduli space.

Connects algebraic invariants with geometric height calculations.

Abstract

Let $L$ be a Galois algebra with Galois group $G$ and let $x$ be a normal element of $L$. The moduli space $\mathcal X$ of pairs $(L,x)$ is isomorphic to an open subset of the quotient variety $\mathbb P/G$, where $\mathbb P$ is the projective space of the regular representation of $G$. We provide a formula for the height of any pair $(L,x) \in \mathcal X(\mathbb Q)$ in terms of algebraic invariants of $L$ and $x$ with respect to a natural adelic metric on the anticanonical divisor of $\mathbb P/G$.