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I am trying to understand irreducibility of polynomials in 2 variables. Specifically, I want to prove that if $f(t)$ is irreducible in $F[t]$ (where $F$ is a field), then $f(s-t)$ is irreducible in $F(t)[s]$.

Is there a criterion\theorem which helps to prove this result? Is it even true?

It feels to me like an opposite direction of Hilbert's irreducibility theorem, i.e. turning an irreducible element of a polynomial ring with 1 variable, into an irreducible element of a polynomial ring with 2 variables. I hope it is not hard, I actually feel it is easy, but I don't have enough experience with polynomial rings over 2 variables.

EDIT: What about the following generalization? If $f(t)$ is irreducible in $F[t]$ and $h(s,t)$ is irreducible in $F[s,t]$, is $f(h(s,t))$ irreducible in $F(t)[s]$? Is it true?

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It seems to me that since $t$ is in the coefficient field for $f(s-t)$ it can't affect factorization; $f(s-t)$ is irreducible over $F(t)$ if and only if $f(s)$ is. And the factorization of $f(s)$ can't involve the indeterminate $t$; if it's irreducible over $F$, it's still irreducible over $F(t)$.

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Can you explain the last point ("the factorization of f(s) can't involve the indeterminate t")? It sounds intuitive, but when I try to prove it I get many different cases and none of them seems easy or simple. –  Ofir Mar 24 '12 at 14:50
    
OK, I managed to prove it. Gauss's Lemma reduced it to proving that $f(s)$ is irreducible in $F[s][t]$. Then I used the fact that the only invertible polynomials in an integral domain are constant. –  Ofir Mar 24 '12 at 15:58

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