Artin–Schreier theory

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

For a field $F$ of characteristic $p$, the exact sequence $0\to {\mathbb{F}}_p\to F^{\mathrm{sep}}\stackrel{z\mapsto z^p-z}{\to }{F}^{\mathrm{sep}}\to 0$, together with additive normal-basis acyclicity, gives $H^2(\mathrm{Gal}(F^{\mathrm{sep}}/F),{\mathbb{F}}_p)=0$. Consequently every central finite embedding problem with group homomorphism kernel $C_p$ has a continuous weak solution: its pulled-back two-cocycle is a coboundary, whose correcting one-cochain supplies the lift. Over ${\mathbb{F}}_p((t))$, independent Artin–Schreier classes $t^{-m}$, with $m>0$ prime to $p$, allow the lift to be twisted by characters so that inertia maps onto the group homomorphism kernel as well as onto the original quotient. Induction through central order-$p$ quotients therefore realizes every finite p-group as the Galois group of a totally ramified extension with residue field ${\mathbb{F}}_p$.

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 $c\in {\mathbb{F}}_p$.