A note on depth-$b$ normal elements

TL;DR

This paper investigates elements in finite fields with multiple normal elements in a sequence, clarifies previous misconceptions, and generalizes the concept of normal depth to include broader cases.

Contribution

It corrects and extends the definition of normal $eta$-depth, addressing previous errors and answering open questions from earlier work.

Findings

Generalized the definition of normal $eta$-depth.

Clarified the discrepancy in previous results.

Provided new insights into the structure of normal elements.

Abstract

In this paper we study elements $\beta \in \mathbb{F}_{q^n}$ having normal $\alpha$-depth $b$; that is, elements for which $\beta, \beta - \alpha, \ldots, \beta-(b-1)\alpha$ are simultaneously normal elements of $\mathbb{F}_{q^n}$ over $\mathbb{F}_{q}$. In [1], the authors present the definition of normal $1$-depth but mistakenly present results for normal $\alpha$-depth for some fixed normal element $\alpha \in \mathbb{F}_{q^n}$. We explain this discrepancy and generalize the given definition of normal $(1-)$depth from [1] as well as answer some open questions presented in [1].