Let $\alpha = \sqrt[3]{2}e^{\frac{2}{3}i \pi}$. Prove that $x_1 ^2 + \cdots +x_k ^2 + 1 \in \mathbb{Q}[x_1,\ldots,x_k]$ has no roots in $\mathbb{Q}(\alpha)$
I am stuck at this proof. I've succesfully shown that $\mathbb{Q}(\alpha)=\{a+b \alpha+c\alpha^2:a,b,c \in \mathbb{Q} \}$ since the minimal polynomial of $\alpha$ in $\mathbb{Q}$ is $x^3-2$. I also know that $\mathbb{Q}(\alpha) = \mathbb{Q}(\alpha ^2)=\{r+s \alpha^2 +t\alpha^4:r,s,t \in \mathbb{Q} \} $ because of an earlier theorem. My text seems to imply that I should use these facts to prove the statement above, but I don't see how. I can't even prove the case $k=1$.
Could someone please throw a hint? It really doesn't seem that hard to prove, but I'm just not seeing it.