On the Galois Group of the 2-Class Field Towers of Some Imaginary Quadratic Fields

Let $k$ be a number field, $p$ a prime, and k^nr,p the maximal unramified $p$-extension of $k$. Golod and Shafarevich focused the study of k^nr,p/k on Gal(k^nr,p/k). Let $S$ be a set of primes of $k$ (infinite or finite), and k_S the maximal $p$-extension of $k$ unramified outside $S$. Nigel Boston and C.R. Leedham-Green introduced a method that computes a presentation for Gal(k_S/k) in certain cases. Taking $S=\{(1)\}$, Michael Bush used this method to compute possibilities for Gal(k^nr,2/k) for the imaginary quadratic fields $k=\mathbb{Q}(\sqrt{-2379}),\mathbb{Q}(\sqrt{-445}),Q(\sqrt{-1015})$, and $\mathbb{Q}(\sqrt{-1595})$. In the case that $k=\mathbb{Q}(\sqrt{-2379})$, we illustrate a method that reduces the number of Bush's possibilities for Gal(k^nr,2/k) from 8 to 4. In the last 3 cases, we are not able to use the method to isolate Gal(k^nr,2/k). However, the results in the attempt reveal parallels between the possibilities for Gal(k^nr,p/k) for each field. These patterns give rise to a class of group extensions that includes each of the 3 groups. We conjecture subgroup and quotient group properties of these extensions.

Department dc:contributor.department

Mathematics

Year dc:date.issued

2006

Author dc:creator

• Steurer, Aliza Anne

Advisor dc:contributor.advisor

• Washington, Lawrence

Language dc:language.iso

en_US

Handle dc:identifier.uri

http://hdl.handle.net/1903/3740

OAI identifier oai:identifier

oai:drum.lib.umd.edu:1903/3740