Normal bases of small height in Galois number fields

TL;DR

This paper provides an effective version of the normal basis theorem for Galois number fields, establishing explicit bounds on the height of normal bases in terms of degree and discriminant, with improved bounds for prime degrees.

Contribution

It introduces explicit height bounds for normal bases in Galois number fields, enhancing the classical theorem with effective, quantitative results.

Findings

Established explicit height bounds depending on degree and discriminant

Derived improved bounds for prime degree Galois extensions

Provided constructive methods for finding normal bases with bounded height

Abstract

Let $K$ be a number field of degree $d$ so that $K/\mathbb Q$ is a Galois extension. The {\it normal basis theorem} states that $K$ has a $\mathbb Q$-basis consisting of algebraic conjugates, in fact $K$ contains infinitely many such bases. We prove an effective version of this theorem, obtaining a normal basis for $K/\mathbb Q$ of bounded Weil height with an explicit bound in terms of the degree and discriminant of $K$. In the case when $d$ is prime, we obtain a particularly good bound using a different method.