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    Galois group of $x^2+9$

    P-WK45L

    Area
    Algebra
    Topics
    Galois Theory, Splitting Fields
    Status
    draft

    What is the Galois group of \(x^2+9\) over \(\QQ\)?

    Solution

    The roots are \[ \pm 3i, \] so the splitting field is \[ \QQ(i). \] Since \(x^2+1\) is irreducible over \(\QQ\), \[ [\QQ(i):\QQ]=2. \] Thus the Galois group has two elements: the identity and complex conjugation, \[ i\longmapsto -i. \] Therefore \[ \operatorname{Gal}(x^2+9/\QQ)\cong C_2. \]

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