# Artin-Schreier polynomial

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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

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

for α in F. The function ${\displaystyle A:X\mapsto X^{p}-X}$ is p-to-one since ${\displaystyle A(x)=A(x+1)}$. It is in fact ${\displaystyle \mathbf {F} _{p}}$-linear on F as a vector space, with kernel the one-dimensional subspace generated by ${\displaystyle 1_{F}}$, that is, ${\displaystyle \mathbf {F} _{p}}$ 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 ${\displaystyle \mathbf {F} _{p}}$-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 ${\displaystyle F[X]/\langle A_{\alpha }(X)\rangle }$ 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 ${\displaystyle \beta ,~\beta +1,\ldots ,\beta +(p-1)}$.