Show that \[ \QQ(2^{1/3}) \qquad\text{and}\qquad \QQ(\zeta_3 2^{1/3}) \] are isomorphic fields over \(\QQ\), but are not identical as subfields of \(\CC\).
Solution
Let \[ \alpha=2^{1/3}, \qquad \beta=\zeta_3\alpha. \] Both satisfy \[ x^3-2=0. \] The polynomial \(x^3-2\) is irreducible over \(\QQ\) by Eisenstein at \(2\), so it is the minimal polynomial of both \(\alpha\) and \(\beta\). Hence \[ \QQ(\alpha) \cong \QQ[x]/(x^3-2) \cong \QQ(\beta), \] with the isomorphism sending \(\alpha\) to \(\beta\).
The fields are nevertheless distinct as subfields of \(\CC\). Since \(\alpha\) is real, \[ \QQ(\alpha)\subset\RR. \] But \(\beta=\zeta_3\alpha\) is nonreal, so \[ \beta\notin\QQ(\alpha). \] Therefore \[ \QQ(2^{1/3})\ne\QQ(\zeta_3 2^{1/3}) \] as subsets of \(\CC\), even though they are isomorphic as \(\QQ\)-fields.