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  "body": "\n## Statement\n\nLet $F$ be a [[field|0QA8CHH8MG]] and let $C$ be an [[algebraic closure|0MCXX29HWY]] of $F$.\nSuppose that the [[degree|0GS74FGFEK]] $[C : F]$ is finite and greater than $1$.\nThen $[C : F] = 2$, the field $F$ has [[characteristic|1ZB13N78XW]] $0$, $F$ is [[real closed|778KFSM09P]], and $C = F(i)$ with $i^2 = -1$.\n\n## Remarks\n\nThe proof uses Galois theory on the finite extension $C/F$.\nOne first shows that $C/F$ is separable, so that $C/F$ is a finite Galois extension.\nIn characteristic $p$ such an extension of degree greater than $1$ cannot exist, by an argument with the roots of $x^p - x - a$.\nIn characteristic $0$, one shows that $i \\notin F$, that $C = F(i)$, and that $-1$ is not a sum of squares in $F$, by studying cyclic subextensions of prime degree.\nExample: $F = \\R$ and $C = \\C$, where $[\\C : \\R] = 2$, as the [[fundamental theorem of algebra|0JP0W07Y9A]] says.\nWithout the finiteness hypothesis nothing of this kind holds: for $F = \\Q$, the degree $[\\overline{\\Q} : \\Q]$ is infinite.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that the pair $\\R \\subset \\C$ is the only kind of example where a field is a finite step away from being algebraically closed.\nIf finitely many new elements make a field algebraically closed, then one square root of $-1$ is enough, and the field behaves like the real numbers.\n\n### L2 human\n\nIn terms of Galois groups: a field whose absolute Galois group is finite and nontrivial has absolute Galois group of order $2$, and it is real closed.\nSo the only nontrivial finite subgroups of absolute Galois groups have order $2$, and they come from orderings of the field.\nThe theorem is part of the theory of real closed fields developed by Artin and Schreier.\nThat theory also shows that a field can be ordered if and only if it is [[formally real|X32TSWJSYT]].\n",
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