The Scheme of Monogenic Generators I: Representability

TL;DR

This paper introduces a moduli space scheme for understanding when a number ring extension is generated by a single element, connecting it to Hilbert schemes and providing explicit examples.

Contribution

It constructs a scheme parameterizing generators of algebra extensions, offering a new moduli-theoretic approach to monogenicity in number rings.

Findings

Defined the scheme al M_{B/A} for generators

Connected the scheme to Hilbert schemes and configuration spaces

Provided explicit equations and examples

Abstract

This is the first in a series of two papers that study monogenicity of number rings from a moduli-theoretic perspective. Given an extension of algebras $B/A$, when is $B$ generated by a single element $\theta \in B$ over $A$? In this paper, we show there is a scheme $\mathcal{M}_{B/A}$ parameterizing the choice of a generator $\theta \in B$, a "moduli space" of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples.