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Math Help - poly

  1. #1
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    poly

    Hi

    how can I find the basis of [L:Q] for \sqrt {11+5\sqrt{11}}

    thank you
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by jerad View Post
    Hi

    how can I find the basis of  [L:\mathbb{Q}] for  \sqrt {11+5\sqrt{11}}

    thank you
    Let  \alpha=\sqrt{11+5\sqrt{11}} and we have  L=\mathbb{Q}(\alpha) .

     m_{\alpha,\mathbb{Q}}(x) = x^4-22x^2-154 (ask if you want to see how I got this).

    Thus  [L:\mathbb{Q}]=4 , so our basis is  \{1,\alpha,\alpha^2,\alpha^3\} = \{1,\alpha,\sqrt{11},\alpha^3\} .
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  3. #3
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    Hi

    Thank you for your help.

    But I posted the question wrong. I need to find the minimum polynomial for
    \sqrt {11+5\sqrt{11}}. Then I am finding the zeros of the minimum poly. From that I find the splitting field as [L:Q]=[Q(alpha,beta):Q).I got 16 is that correct? And then I don't know how to find the basis for [L:Q]. If you can give me some ideas that would be great.

    Thank you
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  4. #4
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    Hi

    I posted the question wrong. I need to find the minimum polynomial for
    \sqrt {11+5\sqrt{11}}. Then I am finding the zeros of the minimum poly. From that I find the splitting field as [L:Q]=[Q(alpha,beta):Q).I got 16 is that correct? And then I don't know how to find the basis for [L:Q]. If you can give me some ideas that would be great.

    Thank you
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by jerad View Post
    Hi

    I posted the question wrong. I need to find the minimum polynomial for
    \sqrt {11+5\sqrt{11}}. Then I am finding the zeros of the minimum poly. From that I find the splitting field as [L:Q]=[Q(alpha,beta):Q).I got 16 is that correct? And then I don't know how to find the basis for [L:Q]. If you can give me some ideas that would be great.

    Thank you
    I posted the minimum polynomial in my last post.
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  6. #6
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    yes I also found the minimum polynomial and I worked out the zeros of the minimum polynomial. The zeros cojugate.alpha=\sqrt {11+5\sqrt{11}}and beta=\sqrt {11-5\sqrt{11}} .
    and I got [L:Q]=16.Is that correct? I don't really know how to fine the basis of Q(\alpha ,\beta ).
    Could you help with this.

    Thank you
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