Exercise 5, § 2

Let $K$ be a field, $f$ a place of $K$ with values in a maximal ordered field $L$.**

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

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