- Is \(\QQ(2^{1\over 3})/\QQ\) normal?
Solution
No. Let \(\alpha=2^{1/3}\). Its minimal polynomial over \(\mathbb Q\) is \[ x^3-2, \] which is irreducible by Eisenstein at \(2\). Its three roots are \[ \alpha,\quad \omega\alpha,\quad \omega^2\alpha, \] where \(\omega\) is a primitive cube root of unity. Since \(\mathbb Q(\alpha)\subset\mathbb R\), it does not contain the two nonreal roots \(\omega\alpha\) and \(\omega^2\alpha\). Hence the minimal polynomial of \(\alpha\) does not split over \(\mathbb Q(\alpha)\), so \[ \mathbb Q(2^{1/3})/\mathbb Q \] is not normal.