Monogenic Quartic Polynomials and Their Galois Groups

TL;DR

This work uses the Kappe–Warren classification of quartic Galois groups to construct infinite one-parameter families of monogenic quartic polynomials with prescribed Galois groups $\{4T2,4T3,4T4,4T5\}$, each generating a distinct quartic field. For each type, explicit polynomials $f_t(x)$ in a single integer parameter $t$ are given, with irreducibility established and monogenicity characterized by squarefree conditions on associated polynomials in $t$; Dedekind’s index criterion and discriminant analysis underpin these results. The authors prove the families are new by comparing with known explicit families and show the cyclic case $4T1$ remains open. They also provide a detailed discussion of the cyclic case, including existing monogenic cyclic quartic fields and examples, but do not establish an infinite 4T1-family. Overall, the paper advances the construction of infinite, monogenic quartic fields with prescribed Galois groups and clarifies the status of the cyclic case within this program.

Abstract

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. In this article, we use the classification of the Galois groups of quartic polynomials, due to Kappe and Warren, to investigate the existence of infinite collections of monogenic quartic polynomials having a prescribed Galois group, such that each member of the collection generates a distinct quartic field. With the exception of the cyclic case, we provide such an infinite single-parameter collection for each possible Galois group. We believe these examples are new, and we provide evidence to support this belief by showing that they are distinct from other infinite collections in the current literature. Finally, we devote a separate section to a discussion concerning, what we believe to be, the still-unresolved cyclic case.

Theorems & Definitions (5)

• Theorem 1.1
• Theorem 2.1
• Remark 2.2
• Theorem 2.3: Dedekind Cohen
• proof