  Let $k$ be an arbitrary field. We study a general method to solve the
subfield problem of generic polynomials for the symmetric groups over $k$ via
Tschirnhausen transformation. Based on the general result in the former part,
we give an explicit solution to the field isomorphism problem and the subfield
problem of cubic generic polynomials for $\frak{S}_3$ and $C_3$ over $k$. As an
application of the cubic case, we also give several sextic generic polynomials
over $k$.
