I'm studying field theory (I'm at the very begining) and this is one of the exercices in the first chapter about field extensions.
Let $\omega = e^{\frac{2\pi i}{3}}$. Show that the equation $x^{2} + y^{2} = -1$ has no solutions in the field $\mathbb{Q}(\omega 2^{\frac{1}{3}})$.
I have no idea how to work this out. I would be grateful if you can help me.
To elaborate a bit on the hint in the comment.
We have the following isomorphism $\mathbb{Q}(\omega 2^{\frac{1}{3}}) \simeq \mathbb{Q}(2^{\frac{1}{3}})$ so solutions in one field are in bijection with solutions in another field.
But a value $x^2$ or $y^2$ in $\mathbb{Q}(2^{\frac{1}{3}})$ is $\ge 0$, so $x^2 + y^2 \ge 0$ and can never $= -1$. By the isomorphism this also proves there are no solutions in $\mathbb{Q}(\omega 2^{\frac{1}{3}})$.