Proving No Solution in Extension Field

1. May 18, 2013

Szichedelic

1. The problem statement, all variables and given/known data
Let $\beta=\omega\sqrt[3]{2}$, where $\omega=e^{2\pi i/3}$, and let $K=\mathbb{Q}(\beta)$. Prove that the equation $x^{2}_{1}+\cdots+x^{2}_{k}=-1$ has no solution with $x_{i}$ in $K$.

2. Relevant equations

The irreducible polynomial f is the monic polynomial of lowest degree in F[x} that has an element $\alpha$ of K as a root.

3. The attempt at a solution
My first idea is to move the one over and then view f(x)+1 as a polynomial in F[x]. Then, I want to show that there is no element in K which is a root to this polynomial... I believe this would show that there is no solution to the above equation with $x_{i}$ in $K$. However, I'm not sure how to show this for an arbitrary element of K and if this is even the right approach. I was also wondering if it had to do with the fact that $\beta$ contains a cube root, which would mean we would need at least a degree 3 polynomial with coefficients in the rationals... We only have a degree 2.

Last edited: May 18, 2013