In the first three parts of this series, we considered quadratic, cubic and quartic rings (., rings free of ranks 2, 3, and 4 over Z) respectively, and found that various algebraic structures involving these rings could be completely parametrized by the integer orbits of an appropriate group representation on a vector space. These orbit results are summarized in Table 1. In particular, the theories behind the parametrizations of quadratic, cubic, and quartic rings, noted in items #2, 9, and 13 of Table 1, were seen to closely parallel the classical developments of the solutions to the quadratic,.
