Date of Award
1992
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
First Advisor
Jurgen Hurrelbrink
Abstract
In a series of papers published in the 1930's, L. Redei and H. Reichardt established a method for determining the 4-rank of the narrow ideal class group of a quadratic number field, essentially by finding the rank over ${\bf F}\sb2$ of a $\{0,1\}$-matrix determined by applying the Kronecker symbol to the prime divisors of the field discriminant. When this field is of the form ${\bf Q}(m\sp{1\over2}),$ with $m={\pm}p\sb1{\cdots}p\sb{n},$ each $p\sb{i}\equiv3$ (mod 4) prime, this matrix takes a form similar to that of the adjacency matrix for a tournament graph. Also, in this case we can find the 4-rank of the ideal class group in the ordinary sense. In Chapter 1, we introduce this method. In Chapter 2, we explain in detail the equivalence of Redei's method to a more modern one, and give some useful results concerning the ranks of anti-symmetric matrices over ${\bf F}\sb2.$ In Chapter 3, we use matrices to give precise ranges for the 4-rank of the ideal class group in our situation, establish conditions for maximal 4-rank, and give a partial verification of a previous result relating the 4-rank of a real and corresponding imaginary quadratic extension. We conclude with some results concerning circulant tournaments and their matrices.
Recommended Citation
Kingan, Robert John, "Tournaments and Ideal Class Groups." (1992). LSU Historical Dissertations and Theses. 5323.
https://repository.lsu.edu/gradschool_disstheses/5323
Pages
74
DOI
10.31390/gradschool_disstheses.5323
