Artin–Schreier theory

Artin–Schreier theory describes cyclic extensions of prime degree $p$ in characteristic $p$ using equations $x^p-x=a$.

An Artin–Schreier polynomial over a field of characteristic $p$ has the formIf it has one root $\alpha$, all its roots are $\alpha +c$ for $c\in {\mathbb{F}}_p$. It either has a root in the base field and splits completely, or is an irreducible polynomial of degree $p$.

An Artin–Schreier extension is a field extension generated by a root $\alpha$ of an irreducible Artin–Schreier polynomial. It is a cyclic Galois extension of degree $p$, with automorphisms $\alpha \mapsto \alpha +c$ for $c\in {\mathbb{F}}_p$.