Hi
how can I find the basis of [L:Q] for \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
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