I am trying to figure out the logic behind this worked out example in a textbook.

The definition given of the Galois group, Gal(F/K) is the set of all automorphisms of F that fix K. It then proves if F is the splitting field of f(x) over K, then these automorphisms must permute the roots of f(x).

The example it gives is this: $\displaystyle f(x) = x^4-5x^2+6 = (x^2 - 3)(x^2-2)$. Now $\displaystyle F = \mathbb{Q}(\sqrt(2),\sqrt(3))$ is the splitting field of $\displaystyle f(x)$ over $\displaystyle \mathbb{Q}$.

Let $\displaystyle \theta \in Gal(F/\mathbb{Q})$. Then (according to the book) we must have $\displaystyle \theta(\sqrt(2)) = \pm \sqrt(2)$ and $\displaystyle \theta(\sqrt(3)) = \pm \sqrt(3)$. It then goes on to define the automorphisms, which I agree taking the above for granted.

However, what I can't figure out is why $\displaystyle \theta(\sqrt(2)) = \sqrt(3)$ is not a valid permutation? It doesn't violate the definition given in the book. And it doesn't prove anywhere that the automorphisms only permute to other roots in the same irreducible factor. And I can't think of why, just based on the fact of $\displaystyle \theta$ being an isoomorphism that I can elinimate such possibilites as well. Am I missing something obvious?

Thanks in advanace.