Monogenic Quartic Polynomials and Their Galois Groups

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 $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. 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.