Residue computations in rings of algebraic integers

Recent work has focused on doing residue computations that use quantization within a particular dense ring of integers in the complex plane. That work is generalized, and it is shown that a class of cubic integers provides a more efficient and less costly solution than other rings of integers which have been considered previously. In addition, it is shown that certain quartic integer rings provide a simple solution to the problem of approximating roots of unity by using fields of lower degree. In addition, algorithms for approximating real and complex numbers with specific integer rings are developed