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  "body": "\n## Definition\n\nLet $F$ be a [[field|0QA8CHH8MG]].\nThe field $F$ is *formally real* if $-1$ is not a sum of squares in $F$.\nThat is, there are no $n \\ge 1$ and $a_1, \\dots, a_n \\in F$ with $a_1^2 + \\cdots + a_n^2 = -1$.\n\n## Remarks\n\nEquivalently, for all $n \\ge 1$ and $a_1, \\dots, a_n \\in F$, if $a_1^2 + \\cdots + a_n^2 = 0$, then $a_1 = \\cdots = a_n = 0$.\nBy the Artin–Schreier theorem on orderings, $F$ is formally real if and only if $F$ carries a structure of [[ordered field|010JCV9J4E]].\nA formally real field has [[characteristic|1ZB13N78XW]] $0$, since in characteristic $p$ one has $-1 = 1^2 + \\cdots + 1^2$ with $p - 1$ terms.\nExample: $\\Q$ is formally real, because a sum of rational squares is at least $0$ in the usual order.\nNon-example: $\\C$ is not formally real, because $-1 = i^2$; this is why [[$\\C$ cannot be ordered|01CJRJKDAA]].\n",
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