Artin-Schreier extension

Definition

Let $K$ be a field of characteristic $p$, and let $a\in K$ be such that the polynomial $x^p-x-a$ is irreducible over $K$. The Artin-Schreier extension corresponding to $a$ is the field

$K[x]/(x^p-x-a)$

which is also equal to the splitting field of $x^p-x-a$ over $K$. It is a Galois extension whose Galois group is a cyclic group of order $p$.

In fact, any Galois extension of degree $p$ over a field of characteristic $p$ can be realized as an Artin-Schreier extension.

Facts in the definition

• Artin-Schreier polynomial is either irreducible or splits into linear factors

Relation with other properties

Weaker properties

• Cyclic extension
• Galois extension