Normal bases of small height in Galois number fields

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.

9 pages; to appear in Research in Number Theory