## LSU Historical Dissertations and Theses

#### Title

Tournaments and Ideal Class Groups.

1992

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

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.

74

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