Effective normal basis theorem

TL;DR

This paper improves the construction of normal basis elements in finite Galois extensions of Q by providing size-controlled elements and estimates of Minkowski's minima, advancing understanding in algebraic number theory.

Contribution

It introduces a method to find normal basis elements with controlled size and offers estimates for Minkowski's minima of ideals in number fields.

Findings

Constructed normal basis elements with bounded size

Provided new bounds for Minkowski's minima

Enhanced previous results by Fukshansky and Jeong

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.