Maximal 2-Extensions of Number Fields With Limited Ramification

Let K be a number field, p a rational prime, and S a finite set of primes of K, none of which lies above p. Let K S be the maximal pro-p extension of K unramified outside S, and let GS = Gal(KS/K). In the case K = Q , p = 2, S a set of two odd primes, we find a presentation of GS given certain conditions on the primes. If the primes are both congruent to 3 modulo 4, G S is semidihedral with order explicitly given. When one prime is congruent to 3, the other congruent to 1 modulo 4, each a quadratic nonresidue of the other, then GS is a modular group with order explicitly given. The first two members of another (conjectural) family of GS are found. The use of computers in determining a presentation for GS for given S is illustrated in two appendices. The final chapter discusses computational approaches to finding candidates for pro-2 groups appearing as G S in the case where K is imaginary quadratic and S = O.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Perry, David Michael

Contributors dc:contributor

• Boston, Nigel

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI9953108

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86986