The Quintic Gauss Sums

Die Gaussschen Summen von Ordnung fuenf

by Talom Leopold Fossi
Doctoral thesis
Date of Examination:2002-10-25
Date of issue:2004-09-14
Advisor:Prof. Dr. Samuel James Patterson
Referee:Prof. Dr. Ulrich Stuhler
Referee:Prof. Dr. Ina Kersten
Referee:Kriete PD Hartje
Persistent Address: http://dx.doi.org/10.53846/goediss-2471

 

 

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Abstract

English

We investigate the possibility of generalization of the Cassel-McGettrick formula for the quintic Gauss sums. We construct a fundamental region for the fifth cyclotomic field. we describe in general the combinatoric of their geometry. The formula obtained so far satisfied a recurrence relation. The Euler relation is proved. We show how to extract canonicaly a root of unity once we have contructed the fundamental region. The numerical computation shows that a generalization of Cassels-McGettrick formula fails. One should actually attempt to modify the shape of the fundamental region to see if there is a new formula. We prove that there are new conjectures involving Gauss sums. This is actually supported by strong computation within a certain range. In the appendix we show that, there are Gauss sums which are rational intgers. we explicitely give a proof and how to find them.
Keywords: cyclotomic fields; cassels conjecture; Gauss sums; Jacobi sums; character sums; residue symbol; fundamental region

Other Languages

Eine moeglische Veralgeneinerung von Casselscher Formel ist in der Fassung nicht moglich. Wir konstruiiren fundamentale Bereiche zyklotomischer Koerper und dessen kombinatorischen Geometrie. Die Eulersche relation is erfuelt. wir zeigen, wie man am besten eine kanonische Einheitswuerzel herausfinden kann. Einige Vermutungen ueber die Gaussschen summen mit solchen kononischen Einheitswuerzel, nach unseren numerische Ausgabe, sind gegeben. Wir haben an Anhang noch einn Beweise ueber die gaussschen Summen die rational sind gegeben.
Schlagwörter: Zyklotomische Koerper; Cassels conjecture; Gausssche Summen; Jacobische Summen; residue symbol
 
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