Artin-Schreier towers of finite fields

Given a prime number p, we consider the tower of finite fields F-p = L-1 subset of L-0 subset of L-1 subset of & ctdot;, where each step corresponds to an Artin-Schreier extension of degree p, so that for i >= 0, L-i = Li-1[c(i)], where c(i) is a root of X-p - X -a(i-1) and a(i-1)= (c(-1)& ctdot;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 O(c(i)) of the given generator ci for Li over Li-1. In particular, for i >= 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 cj modulo L-j-1(x), where 0 <= j <= i if p is odd, and i >= 2 and 1 <= j <= i if p = 2. We also show that for i >= 0, the Gal(L-i/Li-1)-conjugates of ai form a normal basis of L-i over Li-1. In addition, we obtain the minimal polynomial of c(1) over F-p in explicit form. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.