Number-theoretic techniques are very useful in studying the representation theory of finite groups. In this paper we attempt to introduce such techniques into the representation theory of finite-dimensional semisimple involutory Hopf algebras over a number field. We consider Hopf algebras which contain an integral Hopf algebra order, and study the relation between the structure of the Hopf algebra and the number-theoretic properties of its orders. Let H be a finite-dimensional semisimple involutory Hopf algebra. An element /l E H is called a left integral if hfl = .~(h)/l for all h E H. It is known [5, 61 that there exist left integrals in H for which E(A) # 0. Let A be an order in H, and let IT, be the ideal of all left integrals in A. The ideal E(L) gives much information on the structure of H and A. It plays a role similar to that played by the order of the group in discussing the representation theory of a finite group. In fact it always is a divisor of dim H. More specifically, let A* be the dual order to A in H*, and let L* be the ideal of left integrals in A*. We prove that c(L) c(L*) is the ideal generated by dim H. A is a separable algebra if and only if c(L) = R. If B is an order containing A, and M is the ideal of left integrals contained in B, we show that (c(L) c(&Z)l) (B/A) = 0. In particular, if E(L) = c(M), it follows that B = A. We then prove the following generalization of a theorem of Frobenius: the degree of any absolutely irreducible representation of H divides E(L). If H is the group algebra of a group G, and A is the integral group ring, then E(L) = (1 G I), and this gives Frobenius’ theorem. If G has a normal abelian subgroup N, then it is possible to construct an order for which c(L) = ([G : N]), in which case our result gives us a theorem of Ito. This