Date of Award

1988

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

J. Hurrelbrink

Abstract

This is a contribution to the research that is going on in Algebraic Number Theory, relating classical questions on class numbers and units of a number field F to the structure of $K\sb2$($O\sb{F}$), the Milnor K-group $K\sb2$ of the ring of integers. We are interested in number fields F where the 2-primary subgroup of $K\sb2$($O\sb{F}$) is elementary abelian of rank $r\sb1$(F), the number of real embeddings of F. In (C-$H\sb1$) it is proven that the 2-primary subgroup of $K\sb2(O\sb{F}$) is of the above type if and only if the number field has the following properties: (a) the number field has exactly one dyadic prime, (b) its S-class number is odd and (c) it contains S-units with independent signs. Here, the set S consists of all dyadic and all infinite primes of the number field. The purpose of this paper is to examine the existence of number fields of the above type and to examine their properties with respect to the parity of their class number and the containment of units with independent signs. We will mostly restrict our attention to number fields that are totally real. For any given totally real number field F that satisfies the above properties we will prove that there exist infinitely many real quadratic extensions that also have the above properties. The main theorem will be a classification of these quadratic extensions of F into families that all share the same properties with respect to the parity of their class number and the containment of units with independent signs.

Pages

95

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