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.
Recommended Citation
Berger, Ruth Ilse, "Class Numbers and Units of Number Fields E With Elementary Abelian K(2)O(E)." (1988). LSU Historical Dissertations and Theses. 4483.
https://repository.lsu.edu/gradschool_disstheses/4483
Pages
95
DOI
10.31390/gradschool_disstheses.4483