Exercise 7, § 17

Let D be a field with center K of finite rank over K, endowed with a total order compatible with its additive group structure and such that the product of two positive elements is positive. Prove that D is equal to K. (Prove that K has characteristic zero. If $D\not= K$, observe that there exist elements $x$ in D - K such that Trd$_{D/K}(x) = 0$, and apply Exercise 6, d)).