On the Galois Group of the 2-Class Field Towers of Some Imaginary Quadratic Fields
| dc.contributor.advisor | Washington, Lawrence | en_US |
| dc.contributor.author | Steurer, Aliza Anne | en_US |
| dc.contributor.department | Mathematics | en_US |
| dc.contributor.publisher | Digital Repository at the University of Maryland | en_US |
| dc.contributor.publisher | University of Maryland (College Park, Md.) | en_US |
| dc.date.accessioned | 2006-09-12T05:38:26Z | |
| dc.date.available | 2006-09-12T05:38:26Z | |
| dc.date.issued | 2006-06-02 | en_US |
| dc.description.abstract | 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. | en_US |
| dc.format.extent | 370674 bytes | |
| dc.format.extent | 4671 bytes | |
| dc.format.extent | 7141 bytes | |
| dc.format.mimetype | application/pdf | |
| dc.format.mimetype | application/postscript | |
| dc.format.mimetype | application/postscript | |
| dc.identifier.uri | http://hdl.handle.net/1903/3740 | |
| dc.language.iso | en_US | |
| dc.subject.pqcontrolled | Mathematics | en_US |
| dc.subject.pquncontrolled | 2-class | en_US |
| dc.subject.pquncontrolled | class field | en_US |
| dc.subject.pquncontrolled | class group | en_US |
| dc.subject.pquncontrolled | unramified | en_US |
| dc.subject.pquncontrolled | p-group generation algorithm | en_US |
| dc.title | On the Galois Group of the 2-Class Field Towers of Some Imaginary Quadratic Fields | en_US |
| dc.type | Dissertation | en_US |