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Normal bases of small height in Galois number fields

Lenny Fukshansky, Sehun Jeong

TL;DR

This work addresses the problem of obtaining a normal basis for a Galois number field with small height by making the normal-basis theorem effective. The authors blend Pazuki–Widmer height bounds, a polynomial nonvanishing avoidance argument, and lattice techniques to construct explicit normal bases and bound their heights in terms of the degree $d$ and discriminant $|\Delta_K|$; in general, they prove $h(\beta_i) \leq \frac{d^{4d} (d^2-d+2)^{4d-3}}{2} \binom{d-1}{[(d-1)/2]}^2 |\Delta_K|^{(d-1)(4d-3)}$, with a sharper bound $h(\beta_i) \le |\Delta_K|^{1/2}$ when $d$ is an odd prime. The prime-degree case leverages Dubickas's results and Minkowski's theorem to produce an integral normal basis efficiently. These results contribute an explicit, constructive version of the normal-basis theorem with potential computational applications in number theory and arithmetic geometry.

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.

Normal bases of small height in Galois number fields

TL;DR

This work addresses the problem of obtaining a normal basis for a Galois number field with small height by making the normal-basis theorem effective. The authors blend Pazuki–Widmer height bounds, a polynomial nonvanishing avoidance argument, and lattice techniques to construct explicit normal bases and bound their heights in terms of the degree and discriminant ; in general, they prove , with a sharper bound when is an odd prime. The prime-degree case leverages Dubickas's results and Minkowski's theorem to produce an integral normal basis efficiently. These results contribute an explicit, constructive version of the normal-basis theorem with potential computational applications in number theory and arithmetic geometry.

Abstract

Let be a number field of degree so that is a Galois extension. The {\it normal basis theorem} states that has a -basis consisting of algebraic conjugates, in fact contains infinitely many such bases. We prove an effective version of this theorem, obtaining a normal basis for of bounded Weil height with an explicit bound in terms of the degree and discriminant of . In the case when is prime, we obtain a particularly good bound using a different method.
Paper Structure (4 sections, 9 theorems, 45 equations)

This paper contains 4 sections, 9 theorems, 45 equations.

Table of Contents

  1. Introduction and statement of results
  2. Notation and heights
  3. Proof of Theorem \ref{['main1']}
  4. Proof of Theorem \ref{['main2']}

Key Result

Proposition 1.1

Let $K/{\mathbb Q}$ be a quadratic extension. Then there exists a normal basis $\beta_1,\beta_2 \in K$ with for $i=1,2$, where $c(2)$ is as in rupp.

Theorems & Definitions (10)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Remark 3.1
  • Theorem 4.1: dub, Theorem 1