About the irreducibility in $k[X,Y]$ and in $k(Y)[X]$

Question

Let $k$ be a generic field and $k(Y)$ be the field of rational function in the variable $Y$. If $f\in k[X,Y]$ is an irreducible polynomial, is it true that it is irreducible as polynomial in $k(Y)[X]$?

Answer 1

The following equivalence (=some version of Gauss's lemma) answers your question:

For a polynomial $f(X,Y)\in k[X,Y]$ the following are equivalent:
(i) The polynomial $f(X,Y)\in k[X,Y]$ is irreducible in the ring $k[X,Y]$ .
(ii) The polynomial $f(X,Y)$, seen as a polynomial in the indeterminate $X$ with coefficients in $k[Y]$, is primitive and the polynomial $f(X,Y)$ is irreducible in the ring $k(Y)[X]$.

Here "primitive" means that if you write $f(X,Y)=\sum a_i(Y)X^i$, no positive degree polynomial $g(Y)$ divides all the coefficients $a_i(Y)$ 's .