Suppose $\displaystyle E = \mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}) $. Suppose we have:

$\displaystyle \psi_{\sqrt{2}, -\sqrt{2}}: \mathbb{Q}(\sqrt{3}, \sqrt{5}))(\sqrt{2}) \to (\mathbb{Q}(\sqrt{3}, \sqrt{5}))(-\sqrt{2}) $

$\displaystyle \psi_{\sqrt{3}, -\sqrt{3}}: \mathbb{Q}(\sqrt{2}, \sqrt{5}))(\sqrt{3}) \to (\mathbb{Q}(\sqrt{2}, \sqrt{5}))(-\sqrt{3}) $

$\displaystyle \psi_{\sqrt{5}, -\sqrt{5}}: \mathbb{Q}(\sqrt{2}, \sqrt{3}))(\sqrt{5}) \to (\mathbb{Q}(\sqrt{2}, \sqrt{3}))(-\sqrt{5}) $


Let these equal $\displaystyle \tau_2 $, $\displaystyle \tau_3 $ and $\displaystyle \tau_5 $ respectively. Show that these elements have order 2 in the group $\displaystyle G(E/\mathbb{Q}) $. So just square the elements and verify that you get the identity?

What is the subgroup $\displaystyle H $ of $\displaystyle G(E/ \mathbb{Q}) $ generated by the elements?