TITLE: If $F$ is a formally real field then is $F(\alpha)$ formally real?
QUESTION [6 upvotes]: Let us call a field $F$ $\textit{ formally real }$ if $-1$ is not expressible as a sum of squares in $F$. Now suppose $F$ is a formally real field and $f(x)\in F[x]$ be an irreducible polynomial of odd degree and $\alpha $ is a root of $f(x)$. Is it true that $F(\alpha)$ is also formally real ?

REPLY [4 votes]: Yes, and here is a proof. ${} {} {}$