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

$A_{\alpha }(X)=X^{p}-X-\alpha \,$

for α in F. The function $A:X\mapsto X^{p}-X$ is p-to-one since $A(x)=A(x+1)$. It is in fact $\mathbf {F} _{p}$-linear on F as a vector space, with kernel the one-dimensional subspace generated by $1_{F}$, that is, $\mathbf {F} _{p}$ itself.

Suppose that F is finite of characteristic p. The Frobenius map is an automorphism 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 $\mathbf {F} _{p}$-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 $F[X]/\langle A_{\alpha }(X)\rangle$ 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 $\beta ,~\beta +1,\ldots ,\beta +(p-1)$.