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Proving No Solution in Extension Field

  1. May 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Let [itex]\beta=\omega\sqrt[3]{2}[/itex], where [itex]\omega=e^{2\pi i/3}[/itex], and let [itex]K=\mathbb{Q}(\beta)[/itex]. Prove that the equation [itex]x^{2}_{1}+\cdots+x^{2}_{k}=-1[/itex] has no solution with [itex]x_{i}[/itex] in [itex]K[/itex].

    2. Relevant equations

    The irreducible polynomial f is the monic polynomial of lowest degree in F[x} that has an element [itex]\alpha[/itex] 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 [itex]x_{i}[/itex] in [itex]K[/itex]. 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 [itex]\beta[/itex] 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
  2. jcsd
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