Artin-Schreier towers of finite fields

TL;DR

This work extends Artin–Schreier tower analysis over finite fields to arbitrary primes, showing that the multiplicative orders of the tower generators $\{c_i\}$ and their images $\{a_i\}$ largely coincide ($O(c_i)=O(a_i)$) with a single exception, and deriving a product-form expression for the common order in terms of intermediate congruence data. It provides unconditional results on the order structure, identifies a normal-basis property via the Galois action, and derives explicit minimal polynomials, including a concrete irreducible polynomial for $c_1$ over $\mathbb{F}_p$. The methodology combines detailed norm/order analysis with explicit conjugation behavior and matrix-analytic arguments to connect orders, normal bases, and minimal polynomials in the Artin–Schreier tower. These findings improve understanding of primitive- and normal-element behavior in finite-field towers, with potential implications for cryptography and constructions of high-order elements and normal bases.

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.

Theorems & Definitions (48)

• Theorem 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 3.1
• proof