The Cohen-Lenstra heuristics give predictions for the distribution of the class groups of a random quadratic number field. Cohen and Martinet generalized them to predict the distribution of the class groups of random extensions of a fixed base field, but Malle pointed out that these predictions have errors arising from the roots of unity in the base field. We give amended predictions that account for the influence of roots of unity.   Our predictions are based on a result which produces a formula for the distribution of a random finite abelian group given its moments, i.e., the expected number of surjections onto a fixed group. This result is very general and we expect it to have further applications in arithmetic statistics.   This is part one of a series of two talks on joint work, some in progress, with Melanie Matchett Wood. Both talks should be understandable on their own.