# poly

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• May 8th 2010, 12:48 PM
jerad
poly
Hi

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

thank you
• May 8th 2010, 02:38 PM
chiph588@
Quote:

Originally Posted by jerad
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 $\{1,\alpha,\alpha^2,\alpha^3\} = \{1,\alpha,\sqrt{11},\alpha^3\}$.
• May 9th 2010, 01:59 AM
jerad
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
• May 9th 2010, 02:00 AM
jerad
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
• May 9th 2010, 06:10 AM
chiph588@
Quote:

Originally Posted by jerad
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.
• May 9th 2010, 06:32 AM
jerad
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