More specifically, is it possible that two fields are isomorphic yet one of the fields has only the trivial automorphism while the other field has at least one nontrivial automorphism?
I think the answer is no because isomorphic (to me) means "the same" and the number of automorphisms should fall under the category of "sameness".
I think the answer is yes because Q(2^(1/3)) is isomorphic to Q(2^(1/3)*zeta_3) since both of these fields are isomorphic to Q[x]/(x^3 - 2). But Q(2^(1/3)) has no nontrivial automorphisms because where would you map 2^(1/3)? right?
While, Q(2^(1/3)*zeta_3) has an automorphism that maps 2^(1/3)zeta_3 to 2^(1/3)zeta_3^2. right?
Given fields $K_1$ and $K_2$ and an isomorphism $\varphi : K_1 \to K_2$ and a nontrivial automorphism $f : K_1 \to K_1$, then $\varphi \circ f \circ \varphi^{-1} : K_2 \to K_2$ is a nontrivial automorphism.