Let $r$ is prime, $s = r^m$ and $\tau \in \mathbb{F}_s$. How to prove, that if polynomial $x^r - x - \tau \in \mathbb{F}_s[x]$ has a root, hence it decomposes on linear factors.
2026-04-23 15:00:31.1776956431
Decomposition over finite field
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This is a Artin-Schreier equation. If $\alpha$ is a root, then $\alpha+\beta$ is a root for all $\beta\in\Bbb F_r$. So the polynomial has $r$ distinct roots.