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    Classification of groups of order $4$, and which arise as Galois groups over $\mathbb{Q}$

    P-WOCSV

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

    Classify the groups of order \(4\). Which of them occur as Galois groups over \(\QQ\)?

    Solution

    Let \(G\) have order \(4\). Every group of order \(p^2\) is abelian, so \(G\) is abelian. By the classification of finite abelian groups, there are exactly two possibilities: \[ C_4, \qquad C_2\times C_2. \]

    Both occur as Galois groups over \(\QQ\).

    For \(C_4\), take the cyclotomic extension \[ \QQ(\zeta_5)/\QQ. \] Its Galois group is \[ (\ZZ/5\ZZ)^\times\cong C_4. \]

    For \(C_2\times C_2\), take the biquadratic extension \[ \QQ(\sqrt2,\sqrt3)/\QQ. \] The independent sign changes of \(\sqrt2\) and \(\sqrt3\) give \[ \operatorname{Gal}(\QQ(\sqrt2,\sqrt3)/\QQ) \cong C_2\times C_2. \]

    Thus both isomorphism types of groups of order \(4\) arise as Galois groups over \(\QQ\).

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