Search the corpus

    $\QQ(2^{1/3})$ and $\QQ(\zeta_3 2^{1/3})$ are isomorphic but not equal

    E-AMD-CZJRDRNT

    Area
    Algebra
    Topics
    Field Extensions, Splitting Fields
    Status
    draft

    Show that \(\QQ(2^{1\over 3})\) and \(\QQ(\zeta_3 2^{1\over 3})\) are isomorphic fields, but not identical.

    Solution

    Let \[ \alpha=\sqrt[3]{2},\qquad \beta=\zeta_3\sqrt[3]{2}. \] Both are roots of \(x^3-2\), which is irreducible over \(\QQ\) by Eisenstein at \(2\). Hence \[ \QQ(\alpha)\cong \QQ[x]/(x^3-2)\cong \QQ(\beta), \] with the induced isomorphism sending \(\alpha\) to \(\beta\).

    The two subfields of \(\CC\) are nevertheless different. Since \(\alpha\) is real, \[ \QQ(\alpha)\subseteq\RR. \] But \(\beta\) is nonreal, so \[ \beta\in\QQ(\beta)\setminus\QQ(\alpha). \] Therefore \[ \boxed{\QQ(\sqrt[3]{2})\cong\QQ(\zeta_3\sqrt[3]{2})\quad\text{but}\quad \QQ(\sqrt[3]{2})\ne\QQ(\zeta_3\sqrt[3]{2}).} \]

    Source Collections