The Least Prime Number That Splits Completely In S3-Sextic Number Fields

In number theory, an integer n is quadratic residue modulo an odd prime p if n is congruent to a perfect square modulo p. Otherwise, n is is called a quadratic nonresidue. Bounding the least prime quadratic residue and the least quadratic nonresidue are two very classical problems in number theory. These classical problems can be generalized to any number field K by asking for bounds the least for prime that splits completely or does not split completely, respectively, in the ring of integers of K. The goal of this thesis is to bound the least prime that splits completely in certain nonabelian Galois number fields in terms of the discriminant of the number field. The analogous problem for for abelian number fields was recently considered by Pollack, using an old method of Linnik and Vinogradov. Our approach is different and requires estimating a certain sum in two ways using both analytic and algebraic tools. There is a “trivial” bound in this problem, and in this thesis we prove the first known nontrivial estimates for certain families of nonabelian Galois number fields