Exercise 5, § 2

Let K be a field and f a place of K with values in a maximal ordered field L.

(a) For all $a \in K$ show that there exists an extension off, either to $K(\sqrt{\alpha})$ or to $K(\sqrt{-\alpha})$, which is a place with values in L. (Consider two cases according to whether there exists $a \in K$ such that $f(a^2\alpha)$ is neither 0 nor $\infty$ or $f(x^2\alpha)$ is equal to 0 or $\infty$ for all $x \in K$.)

(b) Deduce from (a) that there exist an extension E of K which is a maximal ordered field and an extension off to E which is a place of E with values in L. (Reduce it to the case where K is Euclidean and use Exercise 4 (a).)