The Factor Ring Structure of Quadratic Principal Ideal Domains

Previous authors have classified the possible factor rings of the Gaussian integers and the Eisenstein integers. Here, we extend this classification to the ring of integers of any quadratic number field, provided the ring has unique factorization. In the case of imaginary quadratic fields, the classification has exactly the same flavor as that of the Gaussian integers and Eisenstein integers. For real quadratic fields, the classification is only slightly more complicated.