Effective normal basis theorem

TL;DR

The paper advances the effective normal basis problem for a finite Galois extension $K/\mathbb{Q}$ by proving the existence of an element $\alpha\in O_K$ whose conjugates form a $\mathbb{Q}$-basis with explicit size control: $|\sigma_i(\alpha)|\le n|D_K|^{1/n}$ and $h(\alpha)\le \frac{\ln|D_K|}{n}+\ln n$. The approach combines a geometry-of-numbers framework with Minkowski minima for ideals, and a polynomial-avoidance argument (Gaudron–Rémond) to locate small lattice points avoiding a hypersurface defined by $\Delta(x)$. This yields not only the main normal-basis bound but also a lower bound on the minimal size of a normal basis and an effective primitive-element result with improved constants, improving prior work by Fukshansky–Jeong. Additionally, the authors provide bounds on Minkowski minima of ideals, tying number-field discriminants to lattice-theoretic quantities. Overall, the work delivers sharper, effective size controls for normal-basis elements and contributes to the effective primitive-element theory for number fields.

Abstract

Let K be a finite Galois extension of Q. The normal basis theorem provides an element of K whose conjugates form a Q-basis of K. Here we obtain such an element with controlled size. This improves a recent result by Fukshansky and Jeong. By the way, we estimate Minkowski's minima of ideals of integers of number fields.