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    Galois group of $x^n-1$ over $\QQ$ as a function of $n$

    P-QEFMC

    Area
    Algebra
    Topics
    Abelian Groups, Galois Theory, Roots of Unity
    Status
    draft

    Compute the Galois group of \(x^n-1\) over \(\QQ\) as a function of \(n\).

    Solution

    The roots of \(x^n-1\) are the powers of a primitive \(n\)th root of unity \(\zeta_n\), so the splitting field is \[ \QQ(\zeta_n). \] Every \(\QQ\)-automorphism is determined by the image of \(\zeta_n\), which must again be a primitive \(n\)th root. Hence \[ \sigma_a(\zeta_n)=\zeta_n^a \] for a unique residue class \[ a\in(\ZZ/n\ZZ)^\times. \] Conversely every such \(a\) defines an automorphism. Composition satisfies \[ \sigma_a\sigma_b=\sigma_{ab}. \] Therefore \[ \operatorname{Gal}(\QQ(\zeta_n)/\QQ) \cong (\ZZ/n\ZZ)^\times. \] In particular the group is abelian and has order \(\varphi(n)\).

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