Hilbert's Theorem 90, periodicity, and roots of Artin-Schreier polynomials

Let $E\mathrm{/}F$ be a cyclic field extension of degree $n$, and let $\sigma$ generate the group $\mathrm{G}\mathrm{a}\mathrm{l}(E\mathrm{/}F)$. If ${\mathrm{T}\mathrm{r}}_F^E(y)={\sum }_{i=0}^{n-1}{\sigma }^{i}y=0$, then the additive form of Hilbert’s Theorem 90 asserts that $y=\sigma x-x$ for some $x\in E$. When $E$ has characteristic $p>0$ we prove that $x$ gives rise to a periodic sequence $x_0,x_1,\dots$ which has period $p{n}_p$, where $n_p$ is the largest $p$-power that divides $n$. We also show, if $y$ lies in the finite field ${\mathbb{F}}_{p^n}$, then the roots of a reducible Artin-Schreier polynomial $t^p-t-y$ have the form $x+u$ where $u\in {\mathbb{F}}_p$ and $x={\sum }_{i=0}^{n-1}{\sum }_{j=0}^{i-1}{z}^{p^j}{y}^{p^i}$ for some $z\in {\mathbb{F}}_{p^e}$ with $e=n_p$. Furthermore, the sequence ${\left({\sum }_{j=0}^{i-1}{z}^{p^j}\right)}_{i\ge 0}$ is periodic with period $pe$.