Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Calculating generators of power integral bases in sextic fields with a real quadratic subfield

István Gaál

TL;DR

This paper tackles the problem of finding generators of power integral bases in sextic fields that contain a real quadratic subfield. It develops an efficient algorithm that reduces the problem to a relative index equation and a relative Thue equation over the real quadratic subfield, then lifts relative generators to absolute ones using unit arithmetic and a sieve-based pruning strategy; the key step is solving a degree-9 integer-coefficient polynomial $P(a_2)=J(\gamma)-1$ to certify candidates. The main contributions are (i) a practical framework combining relative Thue equation solving with sieving to handle large search spaces, (ii) explicit procedures to obtain all generators up to equivalence with high probability, and (iii) observations about generators arising from units, including reciprocal-unit symmetry. The approach significantly speeds up computations compared to general index-form methods and provides insight into the structure of generators in sextic fields generated by a unit. This has practical impact for computational number theory tasks involving monogenicity and integral bases in high-degree extensions.

Abstract

We discuss the problem of calculating generators of power integral bases in sextic fields, especially focusing on the case of sextic fields with real quadratic subfields. Our main purpose is to describe an efficient algorithm for calculating generators of power integral bases. We show that appropriately using integer arithmetics speeds up the calculations considerably. Our experiences lead to some interesting general statements on generators of power integral bases in number fields generated by a unit.

Calculating generators of power integral bases in sextic fields with a real quadratic subfield

TL;DR

This paper tackles the problem of finding generators of power integral bases in sextic fields that contain a real quadratic subfield. It develops an efficient algorithm that reduces the problem to a relative index equation and a relative Thue equation over the real quadratic subfield, then lifts relative generators to absolute ones using unit arithmetic and a sieve-based pruning strategy; the key step is solving a degree-9 integer-coefficient polynomial to certify candidates. The main contributions are (i) a practical framework combining relative Thue equation solving with sieving to handle large search spaces, (ii) explicit procedures to obtain all generators up to equivalence with high probability, and (iii) observations about generators arising from units, including reciprocal-unit symmetry. The approach significantly speeds up computations compared to general index-form methods and provides insight into the structure of generators in sextic fields generated by a unit. This has practical impact for computational number theory tasks involving monogenicity and integral bases in high-degree extensions.

Abstract

We discuss the problem of calculating generators of power integral bases in sextic fields, especially focusing on the case of sextic fields with real quadratic subfields. Our main purpose is to describe an efficient algorithm for calculating generators of power integral bases. We show that appropriately using integer arithmetics speeds up the calculations considerably. Our experiences lead to some interesting general statements on generators of power integral bases in number fields generated by a unit.
Paper Structure (4 sections, 2 theorems, 41 equations)

This paper contains 4 sections, 2 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The algorithm
  4. Examples and experiences

Key Result

Lemma 1

A. If $K$ is monogenic, then $K$ is also relative monogenic over $M$. B. All generators of power integral bases of $K$ are of the form where $A\in{\mathbb Z}_M$, $\nu$ is a unit in $M$ and $\gamma_0$ generates a relative power integral basis of $K$ over $M$.

Theorems & Definitions (2)

  • Lemma 1
  • Lemma 2