Interpreting $F[x,y]$ for $F$ a field. First, is it appropriate to write $F[x,y] = (F[x])[y] $? In particular, if $F$ is a field, then we know $F[x]$ is a Euclidean domain. Are there necessary and sufficient conditions for when $F[x,y]$ is also a Euclidean domain? 

Yes it is appropriate to write $F[x,y]$ as $(F[x])[y]\cong(F[y])[x]$, so there is no ambiguity.
$F[x,y]$ is not a PID as it contains the non-principal ideal $(x,y)$. This means in particular that $F[x,y]$ is not euclidian.
