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  "body": "\n## Summary\n\nA formally real field is a [[field|04C5ZBG1WT]] in which $-1$ is not a sum of squares.\nBy a theorem of Artin and Schreier, these are exactly the fields that admit an [[ordering|0P3VCBDEE2]].\nThey are the setting of real algebra, which studies the real numbers through algebraic properties alone.\n\n## Explanations\n\n### L1 human\n\nIn the real numbers, a sum of squares is never negative, so it is never $-1$.\nA formally real field is a field with this one property of the real numbers.\nThe rational numbers and the field $\\R(t)$ of real rational functions are formally real.\nThe complex numbers are not, because $-1 = i^2$.\n",
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