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Math Help - Normal extension

  1. #1
    Newbie dangkhoa's Avatar
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    Normal extension

    I've been asked to decide whether the following extensions are normal
    (a) Q(cube root of 5,i):Q
    (b) Q(5^(1/4),i):Q
    (c) C(t):C(t^3)
    (d) Q(t): Q(t^3)

    This is my attempts

    (a) Let a = 5^(1/3), then a has minimal polynomial m(x) = x^3 -5 which is irreducible over Q by Eisenstein's criterion with p = 5
    i has min poly x^2 +1 over Q(5^(1/3)) and since i does not belong to Q(5^(1/3)) , then it is irreducible over Q(5^(1/3))

    Then by tower law, [Q(cube root of 5,i):Q] = 3*2 = 6
    Also, the polynomial f(x) = x^3 -5 = (x -a)(x-aw)(x-a(w^2)) where w is the cube root of unity.
    Is it correct so far?
    I've done up to here and dont know how to decide whether normal or not.

    Part b is similar to part a

    But bart c and d looks a bit hard. Can you give me some hints on these parts please?


    Thank you very much
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  2. #2
    Senior Member
    Joined
    Nov 2008
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    Quote Originally Posted by dangkhoa View Post
    I've been asked to decide whether the following extensions are normal
    (a) Q(cube root of 5,i):Q
    (b) Q(5^(1/4),i):Q
    (c) C(t):C(t^3)
    (d) Q(t): Q(t^3)

    This is my attempts

    (a) Let a = 5^(1/3), then a has minimal polynomial m(x) = x^3 -5 which is irreducible over Q by Eisenstein's criterion with p = 5
    i has min poly x^2 +1 over Q(5^(1/3)) and since i does not belong to Q(5^(1/3)) , then it is irreducible over Q(5^(1/3))

    Then by tower law, [Q(cube root of 5,i):Q] = 3*2 = 6
    Also, the polynomial f(x) = x^3 -5 = (x -a)(x-aw)(x-a(w^2)) where w is the cube root of unity.
    Is it correct so far?
    I've done up to here and dont know how to decide whether normal or not.

    Part b is similar to part a

    But bart c and d looks a bit hard. Can you give me some hints on these parts please?


    Thank you very much
    If your definition of normal extension is here, (a) is not normal extension. If your w is a non-real cube root of unity, then the splitting field of your f(x)=x^3-5 over \mathbb{Q} is \mathbb{Q}(\sqrt[3]{5},\sqrt{3}i) or \mathbb{Q}(\sqrt[3]{5},\alpha) ( \alpha is a root of x^2+x+1), which is not equal to \mathbb{Q}(\sqrt[3]{5},i).

    For (b), let f(x)=x^4-5. The splitting field of f(x) over Q is to adjoin 4th root of 5 to \mathbb{Q}. Since two primitive fourth roots of unity are {i, -i}, you can check that (b) is the normal extension.

    (c) is the normal extension while (d) is not.

    Let F=\mathbb{C}(t^3) and F'=\mathbb{Q}(t^3)
    Check if x^3 - t^3 \in F[x] ( resp. x^3 - t^3 \in F'[x]) splits completely in \mathbb{C}(t)[x] ( resp. \mathbb{Q}(t)[x] ).
    Last edited by aliceinwonderland; February 26th 2010 at 04:49 PM.
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