# Artin-Schreier polynomial

In field theory, an **Artin-Schreier polynomial** is a polynomial whose roots are used to generate field extensions of prime degree *p* in characteristic *p*.

An Artin-Schreier polynomial over a field *F* is of the form

for α in *F*. The function is *p*-to-one since . It is in fact -linear on *F* as a vector space, with kernel the one-dimensional subspace generated by , that is, itself.

Suppose that *F* is finite of characteristic *p*. The Frobenius map is an automorphism of *F* and so its inverse, the *p*-th root map is defined everywhere, and *p*-th roots do not generate any non-trivial extensions.

If *F* is finite, then *A* is exactly *p*-to-1 and the image of *A* is a -subspace of codimension 1. There is always some element α of *F* not in the image of *A*, and so the corresponding Artin-Schreier polynomial has no root in *F*: it is an irreducible polynomial and the quotient ring is a field which is a degree *p* extension of *F*. Since finite fields of the same order are unique up to isomorphism, we may say that this is "the" degree *p* extension of *F*. As before, both roots of the equation lie in the extension, which is thus a *splitting field* for the equation and hence a Galois extension: in this case the roots are of the form .