Artin-Schreier towers of finite fields

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, L₈=L₈-₁c₈, where cᵢ is a root of Xᵖ-X-a₈-₁ and a₈-₁= (c-₁ c₈-₁) ^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 cᵢ for Lᵢ over L₈-₁. 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 L₉-₁^, 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/L₈-₁) -conjugates of aᵢ form a normal basis of Lᵢ over L₈-₁. In addition, we obtain the minimal polynomial of c₁ over Fₚ in explicit form.