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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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up vote 3 down vote accepted

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.

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Thank you very much! :) – badatmath Apr 3 '11 at 20:41

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