Artin-Schreier polynomial

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

An Artin-Schreier polynomial over 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.

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 therefore 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 $$\beta,~\beta+1, \ldots,\beta+(p-1)$$.