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Math Help - Galois Group

  1. #1
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    Galois Group

    I'm trying to find the Galois Group of the splitting field of x^{8}-3 over Q.

    This polynomial is irreducible, and the splitting field is generated by \sqrt[8]{3}=a and w, any primitive 8th root of unity.

    Here I am a bit uncertain. I believe like Q(a) is an extension of degree 8. And Q(w)=Q(i) is an extension of degree 2. So would Q(a,w) be an extension of degree 16. But this is the exact same as x^{8}-2 over Q, and that can't be correct.
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  2. #2
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    Quote Originally Posted by robeuler View Post
    I'm trying to find the Galois Group of the splitting field of x^{8}-3 over Q.

    This polynomial is irreducible, and the splitting field is generated by \sqrt[8]{3}=a and w, any primitive 8th root of unity.

    Here I am a bit uncertain. I believe like Q(a) is an extension of degree 8. And Q(w)=Q(i) is an extension of degree 2. So would Q(a,w) be an extension of degree 16. But this is the exact same as x^{8}-2 over Q, and that can't be correct.
    You can go here, it may be helpful.
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  3. #3
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    Quote Originally Posted by ThePerfectHacker View Post
    You can go here, it may be helpful.
    so [Q(a,w):Q]=\varphi(3)*16=32?

    I am very confused about how to calculate the Galois group if this is true. I know it is of order 32, but i don't understand how the roots can be permuted.
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