Artin-Schreier towers of finite fields

TL;DR

This paper studies the structure and properties of Artin-Schreier towers of finite fields, focusing on the multiplicative order of generators, Galois conjugates, and explicit minimal polynomials, extending prior work to arbitrary primes.

Contribution

It generalizes previous results on the multiplicative order of generators in Artin-Schreier towers from prime 2 to arbitrary primes and provides explicit minimal polynomials.

Findings

The order of $c_i$ equals the order of $a_i$, except for $p=2$ and $i=1$.

The order of $c_i$ is the product of orders of certain elements modulo subfields.

The Galois conjugates of $a_i$ form a normal basis of $L_i$ over $L_{i-1}$.

Abstract

Given a prime number $p$, we consider the tower of finite fields $F_p=L_{-1}\subset L_0\subset L_1\subset\cdots$, where each step corresponds to an Artin-Schreier extension of degree $p$, so that for $i\geq 0$, $L_{i}=L_{i-1}[c_{i}]$, where $c_i$ is a root of $X^p-X-a_{i-1}$ and $a_{i-1}=(c_{-1}\cdots c_{i-1})^{p-1}$, with $c_{-1}=1$. We extend and strengthen to arbitrary primes prior work of Popovych for $p=2$ on the multiplicative order of the given generator $c_i$ for $L_i$ over $L_{i-1}$. In particular, for $i\geq 0$, we show that $O(c_i)=O(a_i)$, except only when $p=2$ and $i=1$, and that $O(c_i)$ is equal to the product of the orders of $c_j$ modulo $L_{j-1}^\times$, where $0\leq j\leq i$ if $p$ is odd, and $i\geq 2$ and $1\leq j\leq i$ if $p=2$. We also show that for $i\geq 0$, the $\mathrm{Gal}(L_i/L_{i-1})$-conjugates of $a_i$ form a normal basis of $L_i$ over $L_{i-1}$. In addition, we obtain the minimal polynomial of $c_1$ over $F_p$ in explicit form.