Summary: It is well known that, in general, a cyclotomic scheme \(\mathcal C\) on a finite field \(\mathbb{F}\) cannot be characterized up to isomorphism by its intersection numbers. It is shown that the intersection numbers of some scheme \(\widehat{\mathcal C}^{(b)}\) on the \(b\)-fold Cartesian product of \(\mathbb{F}\) (\(b\) is the base number of the group \(\operatorname{Aut}({\mathcal C})\)) form a full set of invariants of \(\mathcal C\). It is important to note that \(b\leq 3\) for a proper \(\mathcal C\) and that the scheme \(\widehat{\mathcal C}^{(b)}\) can be defined for an arbitrary scheme \(\mathcal C\) (not necessarily cyclotomic) in a purely combinatorial way. The proof of the main result is based on a complete description of the normal Cayley rings and normal Schur rings (introduced in this paper) over a finite cyclic group. The technique developed here makes it possible to show that any Schur ring over a cyclic group that is different from the group ring has a nontrivial automorphism.