Artin–Schreier theory

Definition

Let $p$ be a prime and let $K$ be a field of characteristic $p$. The Artin–Schreier map is the additive homomorphism $\wp \colon K \to K$, $\wp(x) = x^p - x$. For $a \in K$, the Artin–Schreier polynomial of $a$ is $x^p - x - a$, and an Artin–Schreier extension of $K$ is a field $K(\alpha)$ with $\alpha^p - \alpha = a$ for some $a \in K$. Artin–Schreier theory is the description of the cyclic extensions of degree $p$ of $K$ by these polynomials:

1. if $a \in \wp(K)$, then $x^p - x - a$ splits into linear factors over $K$;
2. if $a \notin \wp(K)$, then $x^p - x - a$ is irreducible over $K$, and $K(\alpha)/K$ is a Galois extension that is cyclic of degree $p$;
3. every cyclic extension of degree $p$ of $K$ inside a separable closure $K^{s}$ arises as in clause 2;
4. the subgroups of order $p$ of $K/\wp(K)$ correspond bijectively to the cyclic extensions of degree $p$ of $K$ inside $K^{s}$.

In clause 4, the subgroup generated by the class of $a \notin \wp(K)$ corresponds to $K(\alpha) \subseteq K^{s}$, where $\alpha^p - \alpha = a$.