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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!

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Because if $f(x) = q_1(x)\cdots q_n(x)$ is an expression of $f$ as a product of irreducibles, then applying the isomorphism between $F[x]$ and F'[x] you get that f'(x) = q_1'(x)\cdots q_n'(x) is an expression of f'(x) as a product.

That each q_i'(x) is irreducible follows because any expression of q_i'(x) as a product can be "brought back" to $F[x]$ via the inverse of $\varphi$ to get an expression of the irreducible $q_i(x)$ as a product, which is impossible.

Unique factorization now tells you that you have a correspondence between the irreducible factors of $f(x)$ and the irreducible factors of its image.