Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1985 On K(,2) of Rings of Integers of Totally Real Number Fields (Birch-Tate, Steinberg, Class Number, Symbol, Zeta-Function). Karl Friedrich Hettling Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Hettling, Karl Friedrich, "On K(,2) of Rings of Integers of Totally Real Number Fields (Birch-Tate, Steinberg, Class Number, Symbol, Zeta-Function)." (1985). LSU Historical Dissertations and Theses. 4056. https://digitalcommons.lsu.edu/gradschool_disstheses/4056 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. 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University Micrdfilms International 300 N. Zeeb Road Ann Arbor, Ml 48106 8517739 Hettling, Karl Friedrich ON K(,2) OF RINGS OF INTEGERS OF TOTALLY REAL NUMBER FIELDS The Louisiana State University and Agricultural and Mechanical Col.Ph.D. 1985 University Microfilms International300 N. Zeeb Road, Ann Arbor, Ml 48106 On K% of Rings of Integer* of Tbtally Real Number Fields A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Karl Friedrich Hettling B.S., Universitat Hamburg, 1978 M.S., Universitat Hamburg, 1982 May 1985 Contents List of tables ................................................................................................................................... Hi Abstract ........................................................................................................................................... to Introduction ......................................................................................................................................v Chapter 1 Preliminaries .................. 1 §1.1 Basic definitions ................................................................................................................... 1 §1.2 Kg of rings of integers ........................................................... 3 C h ap ter 3 Odd divisors of the order of K^(o)................... 7 §2.1 Computation of fjp (-l) ...............................................................................................7 §2.2 Large divisors of the order off Kg(o) ...................................................................................8 §2.3 Small divisors of the order of Kg(o) ................................................................................. 10 §2.4 Irregular primes and #Cg(o)...............................................................................................12 C hapter 8 The 2-part of K 2(o) and the Birch-Ihte conjecture .....................................16 §3.1 Prerequisites for the proofs .......... 15 §3.2 The Birch-T^te conjecture for two families of abelian number fields ............................ 16 C hapter 4 The 2-part of the Birch-Thte conjecture for not necessarily abelian fields 28 §4.1 The proof .................................. 28 §4.2 Numerical results.............................................................. 32 Bibliography ................................................................................................................... 34 T&blm ............................................................................................................................................86 Table 1 .............................................................................................................. 37 Ihble 2 ........................................................................................................................................ 40 Table 3 ............................ 42 A ppendix A program for the computation of ?jp(—1) ...........................................................48 V ita..................................................................................................................................................49 il List of Tables Tkblo 1 The values of ItwgCF) 1)| for the totally real subfields of Q(?p), P < 100, prime ...........................................................................37 TWble 2 The values of ^ ( F ) for the maxima! totally real subfields of m < 100, composite ............... .................................................40 Thble 8 The prime numbers p < 10000 with the property that q — is prime and 2 is a primitive root of q .................................................... 42 iii Abstract We stud; the finite abelian groups JS^o), where o denotes the ring of integers of a totally reai number field. As a major too! we employ the Bkch-T&te conjecture which states that the order of KgO?) can be computed via the Dedekind zeta-function. The odd part of this conjecture has been proved for abelian fields as a consequence of the Mazur-Wiles work on the “Main conjecture®. After the preliminaries of chapter 1, we proceed in chapter 2 by deriving a formula for 1), where F denotes a totally real abelian number Held. Using this formula we prove the congruence 1 = 1 mod [F : Q] for a class of large prime divisors I of For the totally real subfields of Q(fp), p prime, we obtain that every prime q > 5 dividing the field degr ee [ F : <Q] is a divisor of #Kg(o). Finally we show that a prime number p is Irregular if and only if p divides the order of JC^Ojp), where F is the maximal totally real subfield of <Q(cp). In chapter 3 we use results of J. Hurrelbrink and M. Kolster to prove the 3-part of the Birch-Ih&e conjecture for two families of abelian number fields, one of them being the totally real subfields of fc € IN. We compute the 2-parts of W 2(F )fp (-l) and show that the full cyclotomic fields involved have odd class numbers. In chapter 4 we combine recent results of J. Hurrelbrink and P. E. Conner with those of K. S. Brown on the values of the Dedekind zeta-function and obtain that the conditions 2^ :<^11| #Kg(o) and || 1) are equivalent. Therefore the 2-part of the Birch-Thte conjecture holds for any—not necessarily abelian—totally real number field satisfying one (and hence both) of these conditions. Ifeble 1 and table 2 contain the values of |w 2(F )jjp(-l)| for totally real subfields of Q(fm)> m < 100. In tabie 3 we list all primes p < 10 000 with the property that g = is prime and 2 is a primitive root of q. iv Introduction Having undergone quite a rapid development within the last twenty-odd years, algebraic Jf-theory has found applications in various mathematical disciplines. In the case of number fields F and their rings of integers o the study of the various JC-groups Kq(o), JCj(o), K 3 (o), ... is closely related to the arithmetic of the underlying rings. So it is well known [24] that Kq(o) cxB xC l, where Cl denotes the ideal class group of F, and Kj(o) £2 o*, the unit group of o. For the abelian group K^ip) there are no general structure theorems of thfe kind. A theorem of H. Garland [11] states that Kg(o) is finite. The algebraic meaning of JFC 3 (o) is that it contains the “nontrivial” relations between elementary matrices with entries in o. For rings of integers of number fields it is well known that E(o), the group generated by elementary matrices, is equal to SL(o), the special linear group,