A note on $U_n\times U_m$ modular invariants We consider the rings of invariants $R^G$, where $R$ is the symmetric algebra of a tensor product between two vector spaces over the field $F_p$ and $G=U_n\times U_m$. A polynomial algebra is constructed and these invariants provide Chern classes for the modular cohomology of $U_{n+m}$. Keywords:Invariant theory, cohomology of the unipotent groupCategory:13F20