Ufyukyu

up vote 1 down vote favorite I have a question related to count the number of all $mathbb{Q}$ -automorphisms of $mathbb{Q}(sqrt{3},sqrt{5})$ . Since $mathbb{Q}(sqrt{3},sqrt{5})$ is a splitting field of $(x^2-3)(x^2-5)$ over $mathbb{Q}$ . There is a theorem guarantee that (such theorem require $mathbb{Q}(sqrt{3},sqrt{5})$ to be a splitting field of some polynomial): There is an automorphism maps $sqrt{3}$ to $sqrt{3}$ and fix $Bbb Q$ . There is an automorphism maps $sqrt{3}$ to $-sqrt{3}$ and fix $Bbb Q$ . There is an automorphism maps $sqrt{5}$ to $sqrt{5}$ and fix $Bbb Q$ . There is an automorphism maps $sqrt{5}$ to $-sqrt{5}$ and fix $Bbb Q$ . So there are $2times 2=4$ possible automorphisms, $|textbf{Aut}_{Bbb Q}mathbb{Q}(sqrt{3},sqrt{5})|=4$ . But my question is, is there indeed an automorphism maps $sqrt{3}$