Let $f = 3X^4 + 2X^3Y + X^2Y^2 + 3XY^3 + Y^4$. Let $R =\mathbb{Z}[Y]$, $F = Frac[R]$.
I want to know whether or not $f$ is irreducible in $F[X]$. I've tried using the Eisentstein criterion, but I cannot find any suitable primes. I have also managed to factor it in to $f = (X+Y)(3X^3 - X^2Y + 2XY^2 +Y^3)$. I believe this should help, but I'm confused about the ring $F[X]$. Is it $\mathbb{Q}[X,Y]$? Or is it $\mathbb{Q}(Y)[X]$, where $\mathbb{Q}(Y)$ denotes all the rational coefficient polynomials in $Y$.