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    Normality of $\QQ(\sqrt[3]{2})/\QQ$

    E-AMD-BYJBCLC5

    Area
    Algebra
    Topics
    Field Extensions, Galois Theory, Splitting Fields
    Status
    draft

    Is \(\QQ(2^{1\over 3})/\QQ\) normal?

    Solution

    No. Let \(\alpha=\sqrt[3]{2}\). The polynomial \[ f(x)=x^3-2 \] is irreducible over \(\QQ\) by Eisenstein at \(2\), so it is the minimal polynomial of \(\alpha\) over \(\QQ\).

    Its roots are \[ \alpha,\qquad \alpha\zeta_3,\qquad \alpha\zeta_3^2, \] where \(\zeta_3=e^{2\pi i/3}\). Since \[ \QQ(\alpha)\subseteq\RR, \] the two nonreal roots \(\alpha\zeta_3\) and \(\alpha\zeta_3^2\) do not lie in \(\QQ(\alpha)\).

    Thus an irreducible polynomial over \(\QQ\) has a root in \(\QQ(\alpha)\) but does not split there. Hence \[ \QQ(\sqrt[3]{2})/\QQ \] is not normal.

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