poly

May 2009
15
0
Hi

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

thank you
 

chiph588@

MHF Hall of Honor
Sep 2008
1,163
429
Champaign, Illinois
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 2009
15
0
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 2009
15
0
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
Sep 2008
1,163
429
Champaign, Illinois
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 2009
15
0
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