#### Date of Award

1985

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Abstract

We study the finite abelian groups K(,2)(o), where o denotes the ring of integers of a totally real number field. As a major tool we employ the Birch-Tate conjecture which states that the order of K(,2)(o) 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 (zeta)(,F)(-1), where F denotes a totally real abelian number field. Using this formula we prove the congruence L (TBOND) 1 mod F:/Q for a class of large prime divisors L of #K(,2)(o). For the totally real subfields of /Q((zeta)p), p prime, we obtain that every prime q (GREATERTHEQ) 5 dividing the field degree F:/Q is a divisor of #K(,2)(o). Finally we show that a prime number p is irregular if and only if p divides the order of K(,2)(o(,F)), where F is the maximal totally real subfield of /Q((zeta)p). In chapter 3 we use results of J. Hurrelbrink and M. Kolster to prove the 2-part of the Birch-Tate conjecture for two families of abelian number fields, one of them being the totally real subfields of /Q((zeta)(,3)k), k (ELEM) /N. We compute the 2-parts of (omega)(,2)(F)(zeta)(,F)(-1) 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(' F:/Q )(VBAR)(VBAR)#K(,2)(o) and (,2)(' F:/Q )(VBAR)(VBAR)(omega)(,2)(F)(zeta)(,F)(-1) are equivalent. Therefore the 2-part of the Birch-Tate conjecture holds for any--not necessarily abelian--totally real number field satisfying one (and hence both) of these conditions. Table 1 and table 2 contain the values of (VBAR)(omega)(,2)(F)(zeta)(,F)(-1)(VBAR) for totally real subfields of /Q((zeta)m), m (LESSTHEQ) 100. In table 3 we list all primes p < 10000 with the property that q = (p - 1)/2 is prime and 2 is a primitive root of q.

#### 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

#### Pages

57