---
tag: 7YNMNRGH3T
kind: concept
name: Artin–Schreier theory
msc: [12F10, 12F15]
sources:
  - url: https://en.wikipedia.org/wiki/Artin–Schreier_theory
same_as:
  wikidata: Q713056
  wikipedia: Artin–Schreier theory
---

## Summary

Artin–Schreier theory describes the [[cyclic extensions|VFE5B71E7Y]] of degree $p$ of a field of characteristic $p > 0$.
Each of them is generated by a root of a polynomial $x^p - x - a$.
It plays the role that [[Kummer theory|XZQENYV15Y]] plays in degree $n$ prime to the characteristic, where $x^n - a$ is used instead.

## Explanations

### L1 human

In characteristic $p$, adjoining a $p$-th root of an element $a$ that is not a $p$-th power does not give a Galois extension.
The reason is that $x^p - a$ has only one root, repeated $p$ times.
The fix is the polynomial $x^p - x - a$.
If $\alpha$ is one of its roots, then $\alpha + 1, \alpha + 2, \dots, \alpha + p - 1$ are the others, because $(\alpha + 1)^p = \alpha^p + 1$ in characteristic $p$.
So the symmetries of the extension shift the roots by constants, and they form a cyclic group of order $p$.
Over $\F_2$, the polynomial $x^2 + x + 1$ is of this form, and its root field is $\F_4$.