On the monogenity of quartic number fields defined by $x^4+ax^2+b$

TL;DR

This paper characterizes when quartic fields generated by specific trinomials are monogenic, providing explicit divisibility results and showing uniqueness of generators for power integral bases in many cases.

Contribution

It offers a complete characterization of monogenicity for quartic fields defined by $x^4+ax^2+b$ and determines the highest power dividing the index divisor for primes 2 and 3.

Findings

Characterization of when $Z[eta]$ is integrally closed

Explicit calculation of index divisors for $p=2,3$

Uniqueness of power integral bases for a wide class of such fields

Abstract

For any quartic number field $K$ generated by a root $\alpha$ of an irreducible trinomial of type $x^4+ax^2+b\in Z[x]$, we characterize when $Z[\alpha]$ is integrally closed. Also for $p=2,3$, we explicitly give the highest power of $p$ dividing $i(K)$, the common index divisor of $K$. For a wide class of monogenic trinomials of this type we prove that up to equivalence there is only one generator of power integral bases in $K=Q(\alpha)$. We illustrate our statements with a series of examples.