On the matrices with constant determinant and permanent over roots of unity

Let μm be the group of mth roots of unity. In this paper it is shown that if m is a prime power, then the number of all square matrices (of any order) over μm with non-zero constant determinant or permanent is finite. If m is not a prime power, we construct an infinite family of matrices over μm with determinant one. Also we prove that there is no n×n matrix over μp with vanishing permanent, where p is a prime and n=pα−1.