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

<p>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.</p>

Name thesis:degree_name

M.S. in Mathematics

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2015

Author dc:creator

• Ge, Zhenchao

Contributors dc:contributor

• Micah B. Milinovich
• Thai Hoang Le
• Erwin Mina-Diaz

• Prime Quadratic Residue
• Prime Splits Completely
• Subconvexity
• Mathematics

Repository record dc:identifier

https://egrove.olemiss.edu/etd/670

OAI identifier oai:identifier

oai:egrove.olemiss.edu:etd-1669