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    Groups of order four as Galois groups over $\QQ$

    P-PFUVL

    Area
    Algebra
    Topics
    Abelian Groups, Classification, Galois Theory
    Status
    draft

    Which groups of order \(4\) occur as Galois groups over \(\QQ\)?

    Solution

    Up to isomorphism there are exactly two groups of order \(4\): \[ C_4,\qquad C_2\times C_2. \] Both occur over \(\QQ\).

    1. The cyclic group \(C_4\) occurs.

    The cyclotomic extension \[ \QQ(\zeta_5)/\QQ \] is Galois, and \[ \operatorname{Gal}(\QQ(\zeta_5)/\QQ) \cong (\ZZ/5\ZZ)^\times \cong C_4. \]

    2. The Klein four group occurs.

    The biquadratic extension \[ \QQ(\sqrt2,\sqrt3)/\QQ \] is Galois. Independently changing the signs of \(\sqrt2\) and \(\sqrt3\) gives four automorphisms, so \[ \operatorname{Gal}(\QQ(\sqrt2,\sqrt3)/\QQ) \cong C_2\times C_2. \]

    Thus every group of order \(4\) is realizable as a Galois group over \(\QQ\).

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