$Gal_{\mathbb{Q}}[\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{7}]$ Okay so I'm at a loss. Could someone just give me a general direction as to how to solve this? Much appreciated.
2026-03-29 05:50:21.1774763421
Galois Group of $\mathbb{Q}[\sqrt{3}, \sqrt{5}, \sqrt{7}]$
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You have three different plain-to-see automorphisms of $K=\mathbb Q[\sqrt3,\sqrt5,\sqrt7\,]$:
$\rho:\sqrt3\leftrightarrow-\sqrt3$, $\sigma:\sqrt5\leftrightarrow-\sqrt5$, $\tau:\sqrt7\leftrightarrow-\sqrt7$. Does that help? You do need to show that messing up $\sqrt3$ does not mess $\sqrt5$ up, etc.