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!