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  "body": "\n## Definition\n\nLet $p$ be a prime and let $K$ be a [[field|0QA8CHH8MG]] of [[characteristic|1ZB13N78XW]] $p$.\nThe *Artin–Schreier map* is the additive homomorphism $\\wp \\colon K \\to K$, $\\wp(x) = x^p - x$.\nFor $a \\in K$, the *Artin–Schreier polynomial* of $a$ is $x^p - x - a$, and an *Artin–Schreier extension* of $K$ is a field $K(\\alpha)$ with $\\alpha^p - \\alpha = a$ for some $a \\in K$.\n*Artin–Schreier theory* is the description of the [[cyclic extensions|VFE5B71E7Y]] of degree $p$ of $K$ by these polynomials:\n\n1. if $a \\in \\wp(K)$, then $x^p - x - a$ splits into linear factors over $K$;\n2. if $a \\notin \\wp(K)$, then $x^p - x - a$ is irreducible over $K$, and $K(\\alpha)/K$ is a [[Galois extension|0DS2F7GK05]] that is cyclic of degree $p$;\n3. every cyclic extension of degree $p$ of $K$ inside a [[separable closure|JJ91A1V8NQ]] $K^{s}$ arises as in clause 2;\n4. the subgroups of order $p$ of $K/\\wp(K)$ correspond bijectively to the cyclic extensions of degree $p$ of $K$ inside $K^{s}$.\n\nIn clause 4, the subgroup generated by the class of $a \\notin \\wp(K)$ corresponds to $K(\\alpha) \\subseteq K^{s}$, where $\\alpha^p - \\alpha = a$.\n\n## Remarks\n\nClauses 1 and 2 rest on one fact: if $\\alpha$ is a root, the roots are $\\alpha + i$ for $i \\in \\F_p$, since $(\\alpha + i)^p - (\\alpha + i) = \\alpha^p - \\alpha$.\nExample: over $K = \\F_2$, the map $\\wp(x) = x^2 - x$ sends $0$ and $1$ to $0$, so $a = 1 \\notin \\wp(K)$.\nThen $x^2 + x + 1$ is irreducible over $\\F_2$, and $\\F_2(\\alpha) = \\F_4$ is cyclic of degree $2$ over $\\F_2$.\nNon-example: $x^p - a$ is not an Artin–Schreier polynomial, and over $K = \\F_p(t)$ with $a = t$ it gives a purely inseparable extension, which is not Galois.\nMore generally, the finite subgroups of $K/\\wp(K)$ correspond to the finite abelian extensions of $K$ of exponent $p$.\n",
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