Two Families of Monogenic $S_4$ Quartic Number Fields

TL;DR

This paper identifies specific families of quartic polynomials that generate monogenic $S_4$ number fields, providing explicit criteria and density estimates for such fields using the Montes algorithm.

Contribution

It introduces new criteria for monogeneity in $S_4$ quartic fields and applies the Montes algorithm to establish generators and density bounds.

Findings

Monogenic $S_4$ fields are generated by roots of specific polynomial families.

Explicit square-free conditions ensure monogeneity.

Lower bounds on the density of such fields within certain polynomial families.

Abstract

Consider the integral polynomials $f_{a,b}(x)=x^4+ax+b$ and $g_{c,d}(x)=x^4+cx^3+d$. Suppose $f_{a,b}(x)$ and $g_{c,d}(x)$ are irreducible, $b\mid a$, and the integers $b$, $d$, $256d-27c^4$, and $\dfrac{256b^3-27a^4}{\gcd(256b^3,27a^4)}$ are all square-free. Using the Montes algorithm, we show that a root of $f_{a,b}(x)$ or $g_{c,d}(x)$ defines a monogenic extension of $\mathbb{Q}$ and serves as a generator for a power integral basis of the ring of integers. In fact, we show monogeneity for slightly more general families. Further, we obtain lower bounds on the density of polynomials generating monogenic $S_4$ fields within the families $f_{b,b}(x)$ and $g_{1,d}(x)$.