Finite fields and more generally fields of positive 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 unit i. Also, the p-adic numbers cannot be ordered, since according to Hensel's lemma Q2 contains a square root of −7, thus 12 + 12 + 12 + 22 + )2 = 0, and Qp (p > 2) contains a square root of 1 − p, thus (p − 1)⋅12 + ()2 = 0.The squares of the square roots and are in Q, but are < 0, so that these roots cannot be in Q which means that their expansions are not periodic.
