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? 

Euclidean domains are always principal ideal domains. The ideal $(x,y)$ is never principal. Hence $F[x,y]$ is never a Euclidean domain.
