Artin-Schreier towers of finite fields

Abstract

Given a prime number p, we consider the tower of finite fields Fp = L−1 ⊂ L0 ⊂ L1 ⊂ ··· , where each step corresponds to an Artin-Schreier extension of degree p, so that for i ≥ 0, Li = Li−1[ci], where ci is a root of Xp −X − ai−1 and ai−1 = (c−1 ··· ci−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(ci) of the given generator ci for Li over Li−1. In particular, for i ≥ 0, we show that O(ci) = O(ai), except only when p = 2 and i = 1, and that O(ci) is equal to the product of the orders of cj modulo L× j−1, 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(Li/Li−1)-conjugates of ai form a normal basis of Li over Li−1. In addition, we obtain the minimal polynomial of c1 over Fp in explicit form.