Normal elements in finite fields

TL;DR

This paper presents a straightforward derivation of the formula for counting normal elements in finite field extensions, utilizing the action of units in the Galois group ring.

Contribution

It introduces a simple proof based on group ring units acting transitively on normal elements, clarifying the enumeration process.

Findings

Derived a simple formula for normal elements count

Established the transitive action of Galois group units

Provided a new perspective on normal element enumeration

Abstract

We give a simple derivation of the formula for the number of normal elements in an extension of finite fields. Our proof is based on the fact that units in the Galois group ring of a field extension act simply transitively on normal elements.