Incompleteness of the Giulietti-Ughi arc for large primes

Acknowledgments My appreciation and thanks is owed to many people. First, I would like to thank my adviser and friend Felipe Voloch for his insight, patience and encouragement. I am also greatly indebted to my family and friends for their support and understanding. Finally, a special note of thanks is due to my entire dissertation committee. In this dissertation we show that the Giulietti-Ughi arc is not complete for large primes. This arc is complete for primes which are congruent to three modulo four and less than thirty one. The cardinality of this arc has the same order as the Lunelli-Sce bound. We use two powerful theorems, one on the classifications of Galois groups of quintic polynomials and the other, the ˘ Cebotarev density theorem for function fields to show that there exist points on a certain curve which are not covered by the arc. We then outline a technique which could be used to extend the arc to a complete arc.