Graphs in Number Theory.

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.