Maximal 2-Extensions of Number Fields With Limited Ramification

80 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1999.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.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD