What is the Galois group of \(\mathbb{Q}(\sqrt{2}, \sqrt{3}) / \mathbb{Q}\)?
Solution
Let \[ K=\mathbb Q(\sqrt2,\sqrt3). \] Since \(\sqrt3\notin\mathbb Q(\sqrt2)\), \[ [K:\mathbb Q]=4. \] Moreover, \(K\) is the splitting field of \[ (x^2-2)(x^2-3), \] so \(K/\mathbb Q\) is Galois.
Every \(\mathbb Q\)-automorphism independently chooses the signs of \(\sqrt2\) and \(\sqrt3\): \[ \sqrt2\mapsto\pm\sqrt2, \qquad \sqrt3\mapsto\pm\sqrt3. \] All four sign choices occur, giving four automorphisms. The three nonidentity ones all have order \(2\). Therefore \[ \boxed{\operatorname{Gal}(K/\mathbb Q)\cong C_2\times C_2.} \]