# Characterizing $\operatorname{Gal}(\mathbb{Q}(\sqrt{p_1},\dots,\sqrt{p_n})/\mathbb{Q})$ for $p_i$ primes?

For what $n$ is $$\operatorname{Gal}(\mathbb{Q}(\sqrt{p_1},\dots,\sqrt{p_n})/\mathbb{Q})$$ known, where the $p_i$ are primes?

By Kummer theory, I think that $$\operatorname{Gal}(\mathbb{Q}(\sqrt{p_1},\dots,\sqrt{p_n}))/\mathbb{Q})\simeq A/(\mathbb{Q}^\times)^2$$ where $A$ is the subgroup of $\mathbb{Q}^\times$ generated by $(\mathbb{Q}^\times)^n$ and the $p_i$. But it's not clear to me what $A/(\mathbb{Q}^\times)^n$ is. Intuitively, it should probably be $(\mathbb{Z}/2\mathbb{Z})^n$, since we should have an automorphism permuting $\pm\sqrt{p_i}$ for any choice of indices $i$, and those are the only possible ways to permute the roots. It is somehow obvious that $$A/(\mathbb{Q}^\times)^2\simeq(\mathbb{Z}/2\mathbb{Z})^n$$ if one is thinking just in terms of groups?

• Hint: Induct on the number of primes. – PVAL-inactive Mar 14 '14 at 23:54
• You may find of interest the papers of Mordell and Siegel cited in my answer here. – Bill Dubuque Mar 15 '14 at 0:03