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 \(\displaystyle \{1,\alpha,\alpha^2,\alpha^3\} = \{1,\alpha,\sqrt{11},\alpha^3\} \).