By Erdös-Kac, the distribution of the number of distinct prime factors of a random positive integer up to a large bound X approaches normal distribution with mean and variance log(log(X)). Lemke Oliver and Thorne have shown, using the method of moments, that the same is true for the discriminants of number fields from certain families, namely S_n-fields of degree n, for n at most 5. Results of this kind depend on the availability of counting results, with sufficiently explicit power-saving error terms, for families of number fields subject to local conditions. We discuss families of cyclic extensions for which such counting results, and therefore also Erdös-Kac type results, are available, including the family of cyclic degree-p-fields for every prime p. This is joint work with Marcello Harle-Cowan.​