$-1$ is not a sum of square

Question

Prove that $−1$ is not a sum of squares in the field $\mathbb{Q(\beta)}$ where $\beta = 2^{1/3}e^{2\pi i /3}$

My attempt : In fact $\Bbb Q(\beta)$ and $\Bbb Q(2^{1/3})$ are naturally isomorphic. So I need to show $−1$ is not a sum of squares in the field $\Bbb Q(2^{1/3})$.

Actually I can't find any way to do this after this point. Any help/hint in this regards would be highly appreciated. Thanks in advance!

Answer 1

Disclaimer: I don't know any algebra. Lately I've been "self-teaching" myself a little, based on thinking about MSE questions. My point being this: It took me a few minutes to figure out the point to Ewan Delanoy's hint, while in fact it should have been immediately clear to someone who's been studying the subject, like from books and classes and things:

In fact $\Bbb Q(\beta)$ and $\Bbb Q(2^{1/3})$ are naturally isomorphic. This is just a special case of the following, which must be a basic result that's been covered in class:

Theorem. Suppose $F$ is a field, $\alpha$ and $\beta$ are algebraic and have the same minimal polynomial $m$. Then $F(\alpha)$ and $F(\beta)$ are isomorphic (via $\alpha\mapsto\beta$).

Proof: They're both isomorphic to $F[x]/\langle m\rangle$.