Artins förmodan: p-adiska tal, ändliga kroppar och ekvationer utan heltalslösningar

Abstract

This paper is based on Artin's conjecture concerning homogeneous polynomial equations. The conjecture is false in general but it is still true in many cases. One of our goals is to motivate why the conjecture is formulated the way it is. Moreover, we present a counterproof to the conjecture and we prove the conjecture in one specific case. We construct the p-adic numbers as the conjecture is expressed in terms of p-adic numbers and we introduce theory on finite fields, as it is needed in the motivation of the conjecture, the counterproof and in the proof of the specific case.