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... Show moreIn 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. Show less
We study connections between topological dynamical systems and associated algebras of crossed product type. We derive equivalences between structural properties of a crossed product and dynamical... Show moreWe study connections between topological dynamical systems and associated algebras of crossed product type. We derive equivalences between structural properties of a crossed product and dynamical properties of the associated system and furthermore derive qualitative results concerning the crossed product that are true regardless of the corresponding dynamical system. The systems principally investigated are pairs of a compact Hausdorff space and a homeomorphism, where the integers act on former via iterations of the latter. With such a system a crossed product C*-algebra can be associated. We do not only focus on the C*-crossed product of a system, but also on a Banach *-algebra and a non-complete *-algebra that can both be embedded by *-isomorphisms as dense subalgebras of the C*-algebra; the C*-crossed product is the so-called enveloping C*-algebra of this Banach *-algebra. While investigations of the connections between a system and its C*-algebra have an extensive history, considerations of the other two algebras are new. For these algebras, we derive analogues of results from the case of C*-algebras, but also prove a theorem whose counterpart in the C*-algebra case is false. Furthermore we study the interplay between crossed products of Banach algebras by the integers and naturally associated systems. Show less
