陳其誠臺灣大學：數學研究所周奕志Chou, Yi-ChihYi-ChihChou2007-11-282018-06-282007-11-282018-06-282005http://ntur.lib.ntu.edu.tw//handle/246246/59470本篇論文主要介紹 Buchamnn 的演算法和 Arakelov 類群This thesis focuses not only on Buchmann’s algorithm for computing the class group together with the regulator of an arbitrary number field but also on the basic properties on Arakelov class groups which is relevant to Buchmann’s algorithm.Contents 1 Introduction 2 2 Basic Definitions, Notations and Results 3 2.1 Class groups, fundamental units and regulators 3 2.2 Methods for computing class number and class group 5 2.3 Computations on ideals 7 2.4 Arakelov class group 8 2.5 The idea of Buchmann's algorithm 11 3 Arakelov Class Groups 14 3.1 Inside and outside metrics on Arakelov divisors 14 3.2 The oriented Arakelov class 18 3.3 Reduced Arakelov divisors 20 4 Buchmann’s Algorithm 27 4.1 The LLL-reduced algorithm 27 4.2 Computation on reduced Arakelov divisor 32 4.3 The main steps of Buchmann’s algorithm 34 4.3.1 Finding Ic 34 4.3.2 Looking for relations 34 4.3.3 Producing a generating system 36 4.3.4 Determine the class group and the regulator 37 4.4 The complexity of Buchmann’s algorithm 38 4.5 Application the class group to the principal ideal problem 40380120 bytesapplication/pdfen-USBuchmann演算法Arakelov類群Arakelov class groupregulatorclass groupclass numberthesishttp://ntur.lib.ntu.edu.tw/bitstream/246246/59470/1/ntu-94-R92221018-1.pdf