There is a part of example 10.25.3 at http://stacks.math.columbia.edu/tag/00EX that I'm having trouble understanding. Here, $k$ is a field and $f,g \in k[x,y]$ are irreducible and are not associates. I am confused where they show that $f$ and $g$ are relatively prime when viewed as elements of $k(x)[y]$.
Specifically, why does the following shorter argument not work: $f$ and $g$ are irreducible in $k[x,y] = k[x][y]$ so by Gauss's lemma each $f$ and $g$ are irreducible in $k(x)[y]$. Since $f$ and $g$ are both irreducible, they are, in particular, relatively prime in $k(x)[y]$.
The mere fact that $f$ and $g$ are irreducible in $k(x)[y]$ does not necessarily mean that they are relatively prime. The could still be associates in $k(x)[y]$, say $f = y$ and $g = xy$. You'll need some machinery, using the assumptions that they're irreducible in $k[x,y]$ and not associates in $k[x,y]$, to argue that this can't happen. It looks like that is what is happening in the example you refer to.