Date of Award

1994

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Jurgen Hurrelbrink

Abstract

In the 1930's L. Redei and H. Reichardt established methods for determining the 4-rank of the narrow ideal class group of a quadratic number field, Q($D\sp{1/2}).$ One of these methods involves determining the number of D-splittings of the discriminant, D, of the number field. Later, this method was revised so that we need only find the rank of a matrix over F$\sb2$. In some cases, these Redei matrices can be viewed as adjacency matrices of graphs or digraphs. In Chapter I we introduce the graphs and matrices mentioned above, the method for finding 4-ranks, and present some preliminary results on the number of solutions of $aX\sp{n}$ + $bY\sp{n}$ = 1 over some finite fields. In Chapter II we use the graphs and matrices to study the 4-rank of the ideal class group of Q($D\sp{1/2})$ where D has exactly two prime divisors congruent to 3 modulo 4. In Chapter III, we use graphs to determine the number of solutions of $aX\sp{n}$ + $bY\sp{n}$ = 1 over the finite field Z/pZ up to congruences modulo 2.

Pages

195

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