examples such that (i)$F[x]$ is not a field, (ii) $F[x]$ is also a field. Let $F$ be a field.Then we know that $F[x]$ is a Euclidean Domain.But can someone give me few examples such that (i)$F[x]$ is not a field, (ii) $F[x]$ is also a field.   
Thanks for your kind help.

The polynomial ring F[X] is never a field. Consider the element X. What is it's inverse? It doesn't have one in F[X] (i.e. 1/X isn't in F[X]) and so F[X] can't be field.
