# poly

• May 8th 2010, 12:48 PM
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:

Hi

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

thank you

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

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

Thus $\displaystyle [L:\mathbb{Q}]=4$, so our basis is $\displaystyle \{1,\alpha,\alpha^2,\alpha^3\} = \{1,\alpha,\sqrt{11},\alpha^3\}$.
• May 9th 2010, 01:59 AM
Hi

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
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:

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