A new algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. Using the Chinese remainder theorem or the discrete Fourier transform, this ring can be decomposed into a direct product of fields. This decomposition in turn yields a code construction from codes of lower lengths, which turns out to be in some cases the well-known squaring and cubing constructions, and in other cases the recent \((u+v|u-v)\) and Vandermonde constructions. All binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction. Quinting and septing constructions are introduced. All self-dual quasi-cyclic codes are characterized, and a trace representation is given that generalizes that of cyclic codes.