We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. For a modular categoryC\mathcal {C}withN=ord(T)N= \textrm {ord}(T), the order of the modularTT-matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimensionD2D^2in the Dedekind domainZ[e2πiN]\mathbb {Z}[e^{\frac {2\pi i}{N}}]is identical to that ofNN.