Power integral bases in a family of octic fields

TL;DR

The paper investigates generators of power integral bases in a family of octic fields defined by $f(x)=x^8+ax^6+bx^4+ax^2+1$, showing that beyond the root $\alpha$, nontrivial generators exist. It leverages a quadratic subfield $M$ with $\delta$ to recast the problem as solving a quartic relative Thue equation $P^4-\delta P^2Q^2+Q^4=\varepsilon$ over $M$, using lattice reduction to bound and enumerate small solutions, and then assembling all generators via $\gamma=c_1+\omega c_2+\varepsilon\gamma_0$. A key contribution is the explicit nontrivial generator $\gamma=\left(\frac{a+1}{2}-\omega\right)\alpha-\alpha^3$ when $m=a^2-4b+8$ is square-free, together with a comprehensive computational framework (up to coefficient bound $10^{200}$) for determining all non-equivalent generators in this family. The results illustrate both the existence and structure of multiple generators of power integral bases in these octic fields and provide a practical method for identifying them via relative Thue equations and reduction techniques.

Abstract

Several recent results prove the monogenity of some polynomials. In these cases the root of the polynomial generates a power integral basis in the number field generated by the root. A straightforward question is whether such a number field admits other generators of power integral bases? We have investigated this problem in some previous papers and here we extend this research to a family of octic polynomials, following a recent result of L. Jones.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Lemma 3
• Lemma 4
• Lemma 5
• Theorem 6