Monogenic Quartic Polynomials and Their Galois Groups
Year: 2024
Type: article
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 $f(\theta )=0$ . We use the classification of the Galois groups of quartic polynomials, due to Kappe and Warren [‘An elementary test for the Galois group of a quartic polynomial’, Amer. Math. Monthly96(2) (1989), 133–137], 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 literature. Finally, we devote a separate section to the cyclic case.
Authors: JOSHUA HARRINGTON, LENNY JONES
Institutions: Cedar Crest College, Shippensburg University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,450
OpenAlex ID: W4403341107