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Math Help - splitting field L

  1. #1
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    Question splitting field L

    B is a real number Sqrt(7 + 2*sqrt(7))

    basically i have done the parts of finding the minimal polynomial mu of B over Q, and all the zeros of mu.

    but i got stuck on the part where i need to find the degree [L :Q] and the basis for L over Q

    Where L is the splitting field of mu over Q

    can anyone help me find the degree and basis

    thanks
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  2. #2
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    Quote Originally Posted by dopi View Post
    B is a real number Sqrt(7 + 2*sqrt(7))

    basically i have done the parts of finding the minimal polynomial mu of B over Q, and all the zeros of mu.

    but i got stuck on the part where i need to find the degree [L :Q] and the basis for L over Q

    Where L is the splitting field of mu over Q

    can anyone help me find the degree and basis

    thanks
    Let x = \sqrt{7+2\sqrt{7}} \implies x^2 - 7 = 2\sqrt{7} \implies x^4 - 14x^2+21=0.
    Find the zeros of this polynomial and form the splitting field.
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  3. #3
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    Question check solution

    Quote Originally Posted by ThePerfectHacker View Post
    Let x = \sqrt{7+2\sqrt{7}} \implies x^2 - 7 = 2\sqrt{7} \implies x^4 - 14x^2+21=0.
    Find the zeros of this polynomial and form the splitting field.
    i found the zeros as

    \beta = \pm \sqrt{7 \pm 2\sqrt{7}}
    \alpha= \pm \sqrt{7 \mp 2\sqrt{7}}

    so i tried \alpha\beta, where i got \sqrt{21}, so is the splitting field L of mu over Q =  \sqrt{7}.

    is the degree[L,:Q] = 4x4 = 16
    and the basis for L over Q { {1, \sqrt{7},7, 7\sqrt{7}}}
    and the order of the galois group \Gamma(L:Q) = 8
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  4. #4
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    Quote Originally Posted by dopi View Post
    i found the zeros as

    \beta = \pm \sqrt{7 \pm 2\sqrt{7}}
    \alpha= \pm \sqrt{7 \mp 2\sqrt{7}}

    so i tried \alpha\beta, where i got \sqrt{21}, so is the splitting field L of mu over Q =  \sqrt{7}.

    is the degree[L,:Q] = 4x4 = 16
    and the basis for L over Q { {1, \sqrt{7},7, 7\sqrt{7}}}
    and the order of the galois group \Gamma(L:Q) = 8
    The splitting field is \mathbb{Q}(\sqrt{7 + 2\sqrt{7}},\sqrt{7-2\sqrt{7}}).
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  5. #5
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    Question

    is the degree[L,:Q] = 4x4 = 16
    and the basis for L over Q { {1, \sqrt{7},7, 7\sqrt{7}}}
    and the order of the galois group \Gamma(L:Q) = 8[/QUOTE]

    i was wondering are these correct?
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