The typical elasticity of a quadratic order

It has been common sense to algebraic number theorists that unique factorization may fail in the ring of integers O_K of a certain number field K and that the failure is measured by the class group of K. However, the failure is relatively severe if there exists a nonzero nonunit element of O_K possessing irreducible factorizations of different lengths. This length aspect of the failure can be measured by the “elasticity” of O_K, which is defined as the largest possible ratio of the maximal length to the minimal length of any irreducible factorization occurring in O_K. Pollack recently showed that under GRH, one can take every positive integer as well as every half-integer greater than 1 as the elasticity of infinitely many orders in Q(sqrt{2}). Here an order in a quadratic number field K is a subring of O_K properly containing the ring of rational integers. In this talk, I will discuss the typical size of the elasticity of an order in a fixed quadratic number field, based on joint work with Paul Pollack.