Monogenic Cyclic Quartic Trinomials

TL;DR

This paper classifies all monogenic cyclic quartic trinomials of the form x^4 + bx^2 + d with Galois group cyclic of order 4, proving that only three such polynomials exist.

Contribution

It provides a complete classification of monogenic cyclic quartic trinomials of a specific form with a cyclic Galois group, identifying exactly three such polynomials.

Findings

Exactly three monogenic cyclic quartic trinomials of the form x^4 + bx^2 + d exist with Galois group cyclic of order 4.

Characterization of these trinomials based on their algebraic properties.

Contribution to the understanding of monogenic polynomials with cyclic 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 brief note, we prove that there exist exactly three distinct monogenic trinomials of the form $x^4+bx^2+d$ whose Galois group is the cyclic group of order 4.