If $K$ is a ring consisting only of zero, then $K[x]$ is a field (edit: from the comments below I learned that it's not). Are there another rings with this property? I think no. If $R$ contains $1 \neq 0$, then $1 \cdot x \in R[x]$ has no inverse because a degree of a product of two polynomials is a sum of degrees of the factors.
How do you think?