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arXiv · 2601.04437

Normal bases of small height in Galois number fields

Lenny Fukshansky·Sehun Jeong
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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.

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Lenny Fukshansky, Sehun Jeong. 2026-01-07. Normal bases of small height in Galois number fields. https://arxiv.org/abs/2601.04437

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