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 ?
If $F$ is a formally real field then is $F(\alpha)$ formally real?
6
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abstract-algebra
field-theory
1 Answers
4
Yes, and here is a proof. ${} {} {}$