poly

Hi

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

thank you

chiph588@

MHF Hall of Honor
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 $$\displaystyle \{1,\alpha,\alpha^2,\alpha^3\} = \{1,\alpha,\sqrt{11},\alpha^3\}$$.

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

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

chiph588@

MHF Hall of Honor
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.