Transition to quantum-safe cryptogrpahy is being carried on to resist menace of the developing powerful quantum computers, and lattice-based cryptosystems are among the most primising ciphers in the quantum era to defend confidential communication and also to assure information security. Complete integer lattices are the set of all points with integer coefficients in a linear subspace, and have played a significant rôle in cryptanal-ysis of lattice-based ciphers. This correspondence concentrates on deciding completeness of integer lattices via the local-global principle. Given an integer matrix A consisting of basis vectors of an integer lattice L, two equivalent characterizations of completeness are given, one using ranks of A over residue class fields and the other using decomposition of A over the p-adic ring of integers. From the new characterizations we obtain a new constructive proof of the fact that a complete integer lattice has the same determinant as its orthogonal lattice.