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The 4-Ranks of the TAME Kernels of Quadratic Number Fields

Author: YinXiaoBin
Tutor: QinHouRong
School: Nanjing University
Course: Basic mathematics
Keywords: Number fields TAME nuclear Algebraic K- theory 4 - rank Algebraic number theory
CLC: O156.2
Type: PhD thesis
Year: 2004
Downloads: 47
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Abstract


Algebraic K-theory and algebraic number theory intertwine closely. Let F be a numberfield and OF the ring of the integers in F. Some methods of determining the structure ofthe 2-Sylow subgroups of the tame kernels K2OF have been established, which enable usto verify the famous Birch-Tate conjecture in some cases. Recently, Qin developed in[55]a very effective method of determining the 4-ranks of K2OF for quadratic number fieldsF and then he gave the lower bound of the 4-rank of K2OF for any quadratic numberfield F.In this thesis, we use the method developed in[55] to determine all possible valuesof the 4-ranks of K2OF. In particular, we determine the maximums of the 4-ranks ofK2OF. The thesis is organized as follows.In chapter I, we give a brief introduction to algebraic K-theory and the backgroundof the works on the 2-Sylow subgroups of the tame kernels of number fields.In chapterⅡ, we determine all possible values of the 4-ranks of the tame kernelsK2OF for real quadratic number fields F. In section§2.1, we discuss the method given byQin[55]. In section§2.2, we study the matrix rank over F2. We obtain some interestingresults, which are very useful in determining the maximums of the 4-ranks and whichhave, we think, independent importance. Let F = Q(d1/2), d∈N square-free, be a realnumber field. In sections§2.3 and§2.4, we deal with all possible values of the 4-ranks ofK2OF for d to be odd and even, respectively. We give the upper bounds of the 4-ranksand show that each integer between the lower and the upper bounds occurs as a value ofthe 4-ranks of the tame kernels of infinitely many real quadratic number fields (Theorem2.3.1 and 2.4.1).In chapterⅢ, we establish the similar results as in chapterⅡfor imaginary quadraticnumber fields F(Theorem 3.2.1 and 3.3.1).

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CLC: > Mathematical sciences and chemical > Mathematics > Algebra,number theory, portfolio theory > Number Theory > Algebraic number theory
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