Counting problems for number rings

In this thesis we look at three counting problems connected to orders in number\nfields. First we study the probability that for a random polynomial f in Z[X]\nthe ring Z[X]/f is the maximal order in Q[X]/f. Connected to this is the\nprobability that a random polynomial has a squarefree discriminant.\nThe second counting problem counts the number of subrings within maximal orders.\nWe know that the number of subrings of given index is finite. We determine\nbounds for the number of suborders in terms of the rank of the maximal order and\nthe index of the suborder. Connected to this is a question from Manjul Bhargava\non the number of suborders in quintic rings.\nThe final problem deals with class groups. There are bounds known for the class\nnumber of a maximal order, and we use these bounds to bound the class number of\ngeneral orders.