Universiteit Leiden

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Counting problems for number rings

Promotor: H.W. Lenstra

J.F. Brakenhoff
22 December 2009
Thesis in Leiden Repository

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

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