Infoscience

Thesis

Arithmetic Bounds-Lenstra's Constant and Torsion of K-Groups

This thesis is concerned with computations of bounds for two different arithmetic invariants. In both cases it is done with the intention of proving some algebraic or arithmetic properties for number fields. The first part is devoted to computations of lower bounds for the Lenstra's constant. For a number field K the Lenstra's constant is denoted Λ(K) and defined as the length of the largest exceptional sequence in K. An exceptional sequence is a set of units in K such that for any two among them their difference is a unit as well. H.W. Lenstra showed that if Λ(K) is large enough – bigger than a constant depending on the degree and the discriminant of K – then the ring of integers of K is Euclidean with respect to the norm. Using computer software PARI/GP and some algorithms from graph theory we construct exceptional sequences in number fields having a small discriminant. These exceptional sequences yield lower bounds for Lenstra's constant which are large enough to prove the existence of 42 new Euclidean number fields of degree 8 to 12. The aim of the second part of this thesis is proving upper bounds for the torsion part of the K-groups of a number field ring of integers. A method due to C. Soulé yields bounds for the torsion of these K-groups depending on an invariant of hermitian lattices over number fields. Firstly we describe some properties of rank one hermitian lattices, especially of ideal lattices. Secondly we apply these properties to arbitrary rank hermitian lattices and this implies a significant improvement of the upper bounds for their invariants and accordingly for the torsion of K-groups. The progress mainly achieves much lower contributions of the number field attributes, particularly the degree and the absolute discriminant.

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