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  "body": "\n## Summary\n\nArtin–Schreier theory describes the [[cyclic extensions|VFE5B71E7Y]] of degree $p$ of a field of characteristic $p \u003e 0$.\nEach of them is generated by a root of a polynomial $x^p - x - a$.\nIt plays the role that [[Kummer theory|XZQENYV15Y]] plays in degree $n$ prime to the characteristic, where $x^n - a$ is used instead.\n\n## Explanations\n\n### L1 human\n\nIn characteristic $p$, adjoining a $p$-th root of an element $a$ that is not a $p$-th power does not give a Galois extension.\nThe reason is that $x^p - a$ has only one root, repeated $p$ times.\nThe fix is the polynomial $x^p - x - a$.\nIf $\\alpha$ is one of its roots, then $\\alpha + 1, \\alpha + 2, \\dots, \\alpha + p - 1$ are the others, because $(\\alpha + 1)^p = \\alpha^p + 1$ in characteristic $p$.\nSo the symmetries of the extension shift the roots by constants, and they form a cyclic group of order $p$.\nOver $\\F_2$, the polynomial $x^2 + x + 1$ is of this form, and its root field is $\\F_4$.\n",
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