Existence of normal elements with prescribed norms

TL;DR

This paper investigates the existence of normal elements in finite fields with prescribed norms over intermediate extensions, combining combinatorial and number-theoretic methods to provide both asymptotic and concrete solutions.

Contribution

It introduces the problem of normal elements with prescribed norms over intermediate extensions and provides complete solutions for cases involving a single intermediate extension.

Findings

Solved the problem for one intermediate extension case.

Established asymptotic results for the existence of such normal elements.

Provided concrete examples demonstrating the existence under various conditions.

Abstract

For each positive integer $n$, let $\mathbb F_{q^n}$ be the unique $n$-degree extension of the finite field $\mathbb F_q$ with $q$ elements, where $q$ is a prime power. It is known that for arbitrary $q$ and $n$, there exists an element $\beta\in \mathbb F_{q^n}$ such that its Galois conjugates $\beta, \beta^q, \ldots, \beta^{q^{n-1}}$ form a basis for $\mathbb F_{q^n}$ as an $\mathbb F_q$-vector space. These elements are called normal and they work as additive generators of finite fields. On the other hand, the multiplicative group $\mathbb F_{q^n}^*$ is cyclic and any generator of this group is a primitive element. Many past works have dealt with the existence of primitive and normal elements with specified properties, including the existence of primitive elements whose traces over intermediate extensions are prescribed. Inspired by the latter, in this paper we explore the existence of normal elements whose norms over intermediate extensions are prescribed. We combine combinatorial and number-theoretic ideas and obtain both asymptotic and concrete results. In particular, we completely solve the problem in the case where only one intermediate extension is considered.