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

Abstract

Let $E/F$ be a cyclic field extension of degree $n$, and let $\sigma$ generate the group ${\rm Gal}(E/F)$. If ${\rm Tr}^E_F(y)=\sum_{i=0}^{n-1}\sigma^i y=0$, 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 $pn_p$, 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 $\left(\sum_{j=0}^{i-1}z^{p^j}\right)_{i\ge0}$ is periodic with period $pe$.