How would I prove that the only isomorphisms $\theta :\mathbb{Z[i]} \rightarrow \mathbb{Z[i]}$ are the identity map and $\theta(a+bi) = a-bi$?
I have no idea how to start this, my first thought was to try contradiction, "say there's some other isomorphism $\phi$" but I don't know where to go from there. I'm not asking for a full solution or anything, but any hints or tips would be really appreciated.
Cheers.
Show that a ring automorphism would be determined by where it sends $i$. Where can it?