Artin-Schreier towers of finite fields

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 $i\ge 0$, $L_i=L_{i-1}\left[c_i\right]$, 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 $O\left(c_i\right)$ of the given generator $c_i$ for $L_i$ over $L_{i-1}$. In particular, for $i\ge 0$, we show that $O\left(c_i\right)=O\left(a_i\right)$, except only when $p=2$ and $i=1$, and that $O\left(c_i\right)$ is equal to the product of the orders of $c_j$ modulo $L_{j-1}^{\times }$, where $0\le j\le i$ if p is odd, and $i\ge 2$ and $1\le j\le i$ if $p=2$. We also show that for $i\ge 0$, 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.