Rings of integral quaternions

AbstractA quaternion order derived from an integral ternary quadratic form f = Σaijxixj of discriminant d = 4 det (aij) is m-maximal if m is not divisible by any prime p such that p2 | d, or p ‖; d and cp = 1. If R is m-maximal and m is a product p1 … pr of primes, then any primitive element α of R has unique right-divisor ideals of each norm p1 … pk (k = 1, …, r). This generalizes Lipschitz's ninety-year-old theorem. We characterize m-maximal orders, study their ideals, and show how the preceding result yields formulas for the number of representations of integers by certain quaternary quadratic forms