标题： 希尔伯特第90定理，周期性，以及阿廷-施赖尔多项式的根 显示英文标题

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

摘要： 设$E/F$是一个度数为$n$的循环域扩张，并且让$\sigma$生成群 Gal$(E/F)$。如果 Tr${}^E_F(y)=\sum_{i=0}^{n-1}\sigma^i y=0$，那么希尔伯特第90定理的加性形式断言存在某个$x\in E$使得$y=\sigma x-x$。 假设$E$的特征是$p$。 我们证明了$x$导致一个周期序列$x_0,x_1,\dots$，该序列的周期为$pn_p$，其中 $n_p$是能整除$n$的最大的$p$次幂。 作为应用，我们找到了Artin-Schreier多项式$t^p-t-y$根的闭式表达。 令$y$位于阶为$p^n$的有限域$F_{p^n}$中。Artin-Schreier 多项式$t^p-t-y\in F_{p^n}[t]$恰当在$\sum_{i=0}^{n-1}y^{p^i}=0$时可约。 在这种情况下，$t^p-t-y=\prod_{k=0}^{p-1}(t-x-k)$其中$x=\sum_{i=0}^{n-1}\sum_{j=0}^{i-1}z^{p^j}y^{p^i}$对于某个$z\in F_{p^e}$和$e=n_p$。 序列$\left(\sum_{j=0}^{i-1}z^{p^j}\right)_{i\ge0}$是以$pe$为周期的，并且如果$e$很小，则我们给出显式的$z$。 显示英文摘要

摘要： Let $E/F$ be a cyclic field extension of degree $n$, and let $\sigma$ generate the group Gal$(E/F)$. If 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 $x\in E$. Suppose that $E$ has characteristic $p$. 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 $n$. As an application, we find closed-form expressions for the roots of Artin-Schreier polynomials $t^p-t-y$. Let $y$ lie in the finite field $F_{p^n}$ of order $p^n$. The Artin-Schreier polynomial $t^p-t-y\in F_{p^n}[t]$ is reducible precisely when $\sum_{i=0}^{n-1}y^{p^i}=0$. In this case, $t^p-t-y=\prod_{k=0}^{p-1}(t-x-k)$ where $x=\sum_{i=0}^{n-1}\sum_{j=0}^{i-1}z^{p^j}y^{p^i}$ for some $z\in F_{p^e}$ and $e=n_p$. The sequence $\left(\sum_{j=0}^{i-1}z^{p^j}\right)_{i\ge0}$ is periodic with period $pe$, and if $e$ is small, then we give explicit $z$.