I'm trying to read a proof in Dummit and Foote that says splitting fields of isomorphic fields are isomorphic. There is a passage that goes
"Recall that an isomorphism $\varphi$ from one field $F$ to another field F' induces a natural isomorphism between the polynomial rings $F[x]$ and F'[x]. In particular, if $f(x)$ and f'(x) correspond to one another under this isomorphism then the irreducible factors of $f(x)$ in $F[x]$ correspond to the irreducible factors of f'(x) in F'[x]."
Why is the second sentence true?
Thank you very much!