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Math Help - Galois and isomorphic

  1. #1
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    Galois and isomorphic

    Let L be the splitting field of f(x)=(x^4)-2 over Q. Prove that Gal(L/Q) is a subgroup of S₄ isomorphic to D₄.
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  2. #2
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    Quote Originally Posted by apple2009 View Post
    Let L be the splitting field of f(x)=(x^4)-2 over Q. Prove that Gal(L/Q) is a subgroup of S₄ isomorphic to D₄.
    To show Gal(L/Q) is D_4, you need to verify that the resolvent cubic of f(x) factors as linear times irreducible quadratic and f(x) is irreducible over \mathbb{Q}(\sqrt{D}), where D is the discriminant of the resolvent cubic of f(x). See here or Dummit and Foote's p613-615.
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