Factorization in quadratic fields, ideal class group and L-functions of quadratic fields

FACTORIZATION IN QUADRATIC FIELDS, IDEAL CLASS GROUP AND L-FUNCTIONS OF QUADRATIC FIELDSBülent KÖKLÜCEMathematics, Ph. D. ThesisWhen the ring of integers of a quadratic number field is UFD, then most of the applicationscan be transferred from ordinary integers to this rings which can be done in ordinary numbertheory. But unfortunately the factorization is not always unique for the ring of integers of analgebraic number field as in the case of ordinary integers. The size of non-existence of thisproperty is called the class number. After realizing the failure of unique factorization propertyin most of such rings, number theorists has been expended a great amount of effort toovercome this problem. Such works has given rise to a new field, the ideal theory. Since thenthere has been done so many works on ideal theory and the class number computationproblem. The difficulty of computing the class number has given rise to cryptographicalgorithms based on arithmetic in quadratic fields.In this thesis the factorization and ideal theory in a quadratic field has been studied, widely.The structure and index of ideal class groups is investigated. Three different class numbercomputation method has been studied, two of which are based on algebraic and one is onanalytic methods. The Dirichlet characters, binary quadratic forms, zeta and L-functions ofquadratic number fields has been studied and some results are obtained from this subjects.Quadratic number fields part of the algebraic number fields package of Pari program isintroduced. At the end of each chapter related Pari codes and scripts are given which are usedfor solving some problems and preparing the figures in the thesis.Key Words : Quadratic fields, ring of integers, fundamental unit, Euclidean domain, uniquefactorization domain, ideals, principal ideal domain, ideal class group, binary quadratic forms,reduced forms, class number, Dirichlet characters, zeta functions, L-functions, Pari GP.JURY:1. Assoc. Prof.Dr. Ömer GÖK (Supervisor) Date : 08.12.20052. Prof. Dr. Göksel AĞARGÜN Page : 1303. Assoc. Prof. Dr. Mustafa BAYRAM4. Prof. Dr. Mehmet BAYRAMOĞLU5. Prof. Dr. Elimhan MAHMUDOV