The 4-Ranks Of The TAME Kernels Of Quadratic Number Fields

Algebraic K-theory and algebraic number theory intertwine closely. Let F be a numberfield and O_F the ring of the integers in F. Some methods of determining the structure ofthe 2-Sylow subgroups of the tame kernels K₂O_F 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 K₂O_F for quadratic number fieldsF and then he gave the lower bound of the 4-rank of K₂O_F for any quadratic numberfield F.In this thesis, we use the method developed in[55] to determine all possible valuesof the 4-ranks of K₂O_F. In particular, we determine the maximums of the 4-ranks ofK₂O_F. 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 chapterâ…¡, we determine all possible values of the 4-ranks of the tame kernelsK₂O_F 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(d^1/2), 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 ofK₂O_F 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 chapterâ…¢, we establish the similar results as in chapterâ…¡for imaginary quadraticnumber fields F(Theorem 3.2.1 and 3.3.1).