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

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

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

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

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