Editing Ordered field (section)

==Orderability of fields==
Every ordered field is a [[formally real field]], i.e., 0 cannot be written as a sum of nonzero squares.<ref name=Lam41>Lam (2005) p. 41</ref><ref name=Lam232>Lam (2005) p. 232</ref>

Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. (This order need not be uniquely determined.) The proof uses [[Zorn's lemma]].<ref name=Lam236>Lam (2005) p. 236</ref>

[[Finite field]]s and more generally fields of positive [[Characteristic (algebra)|characteristic]] cannot be turned into ordered fields, as shown above. The [[complex number]]s also cannot be turned into an ordered field, as −1 is a square of the imaginary unit ''i''. Also, the [[p-adic numbers|''p''-adic numbers]] cannot be ordered, since according to [[Hensel's lemma#Examples|Hensel's lemma]] '''Q'''<sub>2</sub> contains a square root of −7, thus 1<sup>2</sup>&nbsp;+&nbsp;1<sup>2</sup>&nbsp;+&nbsp;1<sup>2</sup>&nbsp;+&nbsp;2<sup>2</sup>&nbsp;+&nbsp;{{radic|−7}}<sup>2</sup>&nbsp;=&nbsp;0, and '''Q'''<sub>''p''</sub> (''p''&nbsp;>&nbsp;2) contains a square root of 1&nbsp;−&nbsp;''p'', thus (''p''&nbsp;−&nbsp;1)&sdot;1<sup>2</sup>&nbsp;+&nbsp;{{radic|1&nbsp;−&nbsp;''p''}}<sup>2</sup>&nbsp;=&nbsp;0.<ref>The squares of the square roots {{radic|−7}} and {{radic|1&nbsp;−&nbsp;''p''}} are in '''Q''', but are <&nbsp;0, so that these roots cannot be in '''Q''' which means that their {{nowrap|''p''-adic}} expansions are not periodic.</ref>