On Some Problems Of K₂ Groups

We focus mainly on the K2 of number fields and function fields, and of the ring of algebraic integers, i.e. the tame kernel, which is an important topic in the algebraic K-theory. We consider two problems, one of which is on the p-rank of tame kernel and the other is on the cyclotomic elements in K2 of a field.On the the p-rank of tame kernel, the problem focuses on the relations of the p-ranks between the tame kernel K2OF and the class group of F（ζp）. The first result on the problem is due to Tate. Then, Keune refined Tate’s result. Browkin obtained more precise results in the case of quadratic fields. Following Browkin’s idea, we generalize Browkin’s results for quadratic fields to general number fields. In particular, we discuss the case of cyclic quartic fields in more detail.The problem on the cyclotomic element was first considered by Browkin who try to extend a result of Tate to general number fields, i.e., to write the n-torsion elements in K2（F） as an explicit form when F does not contain the n-th primitive root. On this problem, We modify and change Browkin’s conjecture about cyclotomic elements into more precise forms, in particular we introduce the conception of cyclotomic subgroup. In the rational function field case, we determine completely the exact numbers of cyclotomic elements and cyclotomic subgroups contained in a subgroup generated by finitely many different "linear" cyclotomic elements; while in the number field cases, at first, some number fields F are constructed so that K2（F） contains cyclotomic elements and the cube or square of them are also cyclotomic. These allow us to construct examples of several kind of nontrivial cyclotomic subgroups of order five.In Chapter 1, we give a brief survey on the development of the algebraic K-theory and relative topics.In Chapter 2, we study the relations of the p-ranks between the tame kernel and the ideal class group for a general number field. The main theorems and corollaries are as follows.Theorem 2.1.3 Let F/Q be a Galois extension, E = F（ζp） and F ∩ Q（ζp） = K with l = [K : Q]. Assume that p ≠ l + 1.（i） ef>1,rankp（K2OF）=ranp（ε（（（p-1）/l）-1）Cl（Ep）p）.Moreover,（ii） If ef - 1, then if （Kerλ）∩ε（（（p-1）/l）-1）Cl（E）p does not contain a nontriviai direct summand of ε（（（p-1）/l）-1）Cl（E）p, and otherwise.From this theorem, we get the following corollaries.Corollary 2.1.1 Let F/Q be a Galois extension linearly disjoint from Q（ζp） and E = F（ζp）. If ef > 1, then（i） rankp（K2OF） = rankp（εp₂Cl（E）p）.（ii） rankp（K2OF）=rnkp（η0ε（p-2）Cl（E）p）+rankp(η1ε（p-20 Cl（E）p）.Corollary 2.1.2 Let F/Q be a totally real Calois extension and E = F（ζp）. If （[F: Q],p- 1） = 1, thenIf F is a cyclic quartic field, we have the following theorem:Theorem 2.4.1 Let be a cyclic quartic field and E = F（ζp）.（i） If p D and p > 3, then rankp（K2OF） = rankp（εp-2Cl（E）p）（ii） If pD but p ≠ D, thenTheorem 2.4.3 Let be a quartic field with （i）p=3 then where（ii）If A<0 and 5 h(Q((-D(5+2(5^1/2)))^1/2,then whereTheorem 2.4 be a cyclic quartic field.Then Theorem 2.4.5 Let（i）If（A/5）=1,then（ii）If（A/5）:一1,thenIn Chapter 3,we mainly consider the cyclotomic elements and eyclotomic suUbgroups in K₂ of the rational function field and we get the following results.Theorem 3.4.7 Assume that l≥5 is a prime number and F is a field such that Φl（x）is irreducible in F[x].Let n be an integer saftisfying（i）If ch（F）=0,thell c（Gl（n;F））=2n,and so cs（Gl（n;F））:0.（ii）If ch（F）=p≠0,then c（Gl（n;F））=n（2+|3（l,p）|）.（iii） If eh（F） =p ≠ 0, then we have cs（Gl（n; F）） > 0 （?）l= 3 （rood4） and p is a primitive root of l. In this case, cs（Gt（n; F）） = n, i.e., Gl（n; F） contains exactly n nontrivial eyclotomic subgroups.（iv） Every nontrivial cyclotomic subgroup of Gl（n; F） is a cyclic subgroup of order l, i.e., every nontrivial cyclotomic subgroup has the form Gl（1; F）.Corollary 3.4.6 Assume that l > 5 is a prime number, F is a field with ch（F） ≠ l and Φl（x） is irreducible in Fix]. If n is a positive integer satisfying（i） If eh（F） = 0, then c（G（n; F）） = c(（?）l（n; F） = 2n.（ii） If ch（F） = p ≠ 0, then In particular, when p is a primitive root of I and l --- 3 （rood4）, we haveThe case of n > l-3/2, i.e. l≤ 2n + 1, seems difficult. For n = 2 and l = 5, we have the following theoremTheorem 3.5.1 Assume that F is a field and Φ5（x） is irreducible in F[x].（i） If ch（F） = 0, then c（（?）5（2; F）） = 4, so cs（（?）5（2; F）） = 0.（ii） If eh（F） = p ≠ 0,2, then c（（?）5（2; F）） = 2（2 + |ζ（5,p）|）.We use the symbol Gl*（2; Z） to denote a subgroup of K2（Q（x）） generated by two essentially distinct nontrivial elements of the form where satisfying the ’extra condition’Then we haveTheorem 3.5.2 We have c（G5*（2;Z））= 4, hence cs（G5*（2;Z））= 0, i.e.,G5*（2;Z） contains no nontrivial cyclotomic subgroups.In Chapter 4, we consider the cyclotomic elements and cyclotomic subgroups in Ki of a number field F.Theorem 4.1.1 Assume that p≥3 be a prime. Let a be a zero of the polynomial xp+xp-1+2 and F= Q（α）. Then we have 1≠cp（α）2=cp（a2） ∈ GP（F）.Theorem 4.2.1 Assume that p> 3 is a prime. Let a be a zero of fp,i（x）, where i = 1 or 2, and F= Q（a）. Then we have 1≠cp（α）3= cp（α3） ∈ Gp（F）...