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?