  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.
