Answer to Question #14972 in Calculus for Paul
Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. (This order is often not uniquely determined.)
Finite fields and more generally fields of finite characteristic cannot be turned into ordered fields, because in characteristic p, the element -1 can be written as a sum of (p-1) squares 1². The complex numbers also cannot be turned into an ordered field, as −1 is a square (of the imaginary number i) and would thus be positive. Also, the p-adic numbers cannot be ordered, since Q2 contains a square root of −7 and Qp (p > 2) contains a square root of 1 − p.
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