---
tag: BEEAETX7SB
kind: definition
name: Artin–Schreier theory
msc: [12F10, 12F15]
defines: 7YNMNRGH3T
uses: [1ZB13N78XW, 0QA8CHH8MG, JJ91A1V8NQ, VFE5B71E7Y, 0DS2F7GK05]
sources:
  - url: https://en.wikipedia.org/wiki/Artin–Schreier_theory
---

## Definition

Let $p$ be a prime and let $K$ be a [[field|0QA8CHH8MG]] of [[characteristic|1ZB13N78XW]] $p$.
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|VFE5B71E7Y]] 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|0DS2F7GK05]] that is cyclic of degree $p$;
3. every cyclic extension of degree $p$ of $K$ inside a [[separable closure|JJ91A1V8NQ]] $K^{s}$ 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 $\alpha^p - \alpha = a$.

## Remarks

Clauses 1 and 2 rest on one fact: if $\alpha$ is a root, the roots are $\alpha + i$ for $i \in \F_p$, since $(\alpha + i)^p - (\alpha + i) = \alpha^p - \alpha$.
Example: over $K = \F_2$, the map $\wp(x) = x^2 - x$ sends $0$ and $1$ to $0$, so $a = 1 \notin \wp(K)$.
Then $x^2 + x + 1$ is irreducible over $\F_2$, and $\F_2(\alpha) = \F_4$ is cyclic of degree $2$ over $\F_2$.
Non-example: $x^p - a$ is not an Artin–Schreier polynomial, and over $K = \F_p(t)$ with $a = t$ it gives a purely inseparable extension, which is not Galois.
More generally, the finite subgroups of $K/\wp(K)$ correspond to the finite abelian extensions of $K$ of exponent $p$.