#### Date of Award

2009

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

Sands, Jonathan

#### Abstract

Stark's Conjectures were formulated in the late 1970s and early 1980s. The most general version predicts that the leading coe cient of the Maclaurin series of an Artin L-function should be the product of an algebraic number and a regulator made up of character values and logarithms of absolute values of units. When known, Stark's conjecture provides a factorization of the analytic class number formula of Dirichlet. Stark succeeded in formulating a \re ned abelian" version of his conjecture when the L-function in question has a rst order zero and is associated with an abelian extension of number elds. In the spirit of Stark, Rubin and Popescu formulated analogous \re ned abelian" conjectures for Artin L-Functions which vanish to arbitrary order r at s = 0. These conjectures are identical to Stark's own re ned abelian conjecture when restricted to order of vanishing r = 1. We introduce Popescu's Conjecture C(L=F; S; r): We prove Popescu's Conjecture for multiquadratic extensions when the set of primes S of the base eld is minimal given minor restrictions on the S-class group of the base eld. This extends the results of Sands to the case where #S = r + 1. We present three in nite families of settings where our methods allow us to verify Popescu's conjecture. We formulate a conjecture that predicts when a fundamental unit of a real quadratic eld must become a square in a multiquadratic extension.

#### Recommended Citation

Price, Jason, "Popescu's Conjecture in Multiquadratic Extensions" (2009). *Graduate College Dissertations and Theses*. 187.

http://scholarworks.uvm.edu/graddis/187