Let $GL_n$ be the group of all invertible matrices of order $n$ and $D_n$ the subgroup of $GL_n$ consisting of all invertible diagonal matrices of order $n$. How to show that $D_n$ is closed in $GL_n$ with respect to Zariski topology? I think that we have to show that $D_n$ is the set of zeros of some set of polynomials. What are these polynomials? Thank you very much.