Galois Group of $\mathbb{Q}(\sqrt 2, \sqrt 3)$

Question

Im trying to compute the Galois group of the polynomial $(x^2-2)(x^2-3)$ which has as a splitting field $\mathbb{Q}(\sqrt 2, \sqrt 3)$. The extension is Galois with degree $4$ hence the group has $4$ elements. They should be $\{id,\sigma_1,\sigma_2,\sigma_1\sigma_2\}$,

where $\sigma_1: \sqrt 2 \mapsto - \sqrt 2, \sqrt 3 \mapsto \sqrt 3$ and $\sigma_2: \sqrt 2 \mapsto \sqrt 2, \sqrt 3 \mapsto - \sqrt 3$.

However I'm not understanding why the map $\tau: \sqrt 2 \mapsto \sqrt 3, \sqrt 3 \mapsto \sqrt 2$ is not in the Galois group.

Answer 1

I'm not understanding why the map $\tau: \sqrt 2 \mapsto \sqrt 3, \sqrt 3 \mapsto \sqrt 2$ is not in the Galois group.

Because $\sqrt2^2=2,$ not $3$, so $\tau$ is not an automorphism.