Discriminant and Integral basis of sextic fields defined by x^6+ax+b

TL;DR

This paper investigates the discriminant and integral bases of sextic fields generated by roots of specific degree-six trinomials, providing explicit formulas and methods for constructing integral bases.

Contribution

It offers explicit formulas for the prime power divisors of the discriminant and constructs p-integral bases for sextic fields defined by x^6+ax+b.

Findings

Explicit formulas for discriminant prime divisors.

Method to derive integral bases from p-integral bases.

Illustrative examples demonstrating the methods.

Abstract

Let $K=\mathbb Q(\theta)$ be an algebraic number field with $\theta$ a root of an irreducible trinomial $f(x)=x^6+ax+b$ belonging to $\mathbb{Z}[x]$. In this paper, for each prime number $p$ we compute the highest power of $p$ dividing the discriminant of $K$ in terms of the prime powers dividing $a,~b$ and discriminant of $f(x)$. An explicit $p$-integral basis of $K$ is also given for each prime $p$ and a method is described to obtain an integral basis of $K$ from these $p$-integral bases which is illustrated with examples.