{"id":{"repo_id":"maryland","oai_identifier":"oai:drum.lib.umd.edu:1903/3740"},"canonical_url":"https://search.dev.ndltd.org/etd/maryland/oai:drum.lib.umd.edu:1903/3740","repository":{"repo_id":"maryland","name":"University of Maryland","base_url":"https://api.drum.lib.umd.edu/server/oai/request"},"display":{"title":"On the Galois Group of the 2-Class Field Towers of Some Imaginary Quadratic Fields","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.","abstract_html":"Let $k$ be a number field, $p$ a prime, and <span class=\"etd-inline-math\">k<sup>nr,p</sup></span> the maximal unramified $p$-extension of $k$. Golod and Shafarevich focused the study of <span class=\"etd-inline-math\">k<sup>nr,p</sup>/k</span> on <span class=\"etd-inline-math\">Gal(k<sup>nr,p</sup>/k)</span>. Let $S$ be a set of primes of $k$ (infinite or finite), and <span class=\"etd-inline-math\">k<sub>S</sub></span> the maximal $p$-extension of $k$ unramified outside $S$. Nigel Boston and C.R. Leedham-Green introduced a method that computes a presentation for <span class=\"etd-inline-math\">Gal(k<sub>S</sub>/k)</span> in certain cases. Taking $S=\\{(1)\\}$, Michael Bush used this method to compute possibilities for <span class=\"etd-inline-math\">Gal(k<sup>nr,2</sup>/k)</span> 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&#x27;s possibilities for <span class=\"etd-inline-math\">Gal(k<sup>nr,2</sup>/k)</span> from 8 to 4. In the last 3 cases, we are not able to use the method to isolate <span class=\"etd-inline-math\">Gal(k<sup>nr,2</sup>/k)</span>. However, the results in the attempt reveal parallels between the possibilities for <span class=\"etd-inline-math\">Gal(k<sup>nr,p</sup>/k)</span> 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.","abstract_has_math":true,"creators":["Steurer, Aliza Anne"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":"Mathematics","school":null,"contributors":[],"advisors":["Washington, Lawrence"],"committee_chairs":[],"committee_members":[],"year":2006,"date_issued":"2006-06-02","date_published":"2006-06-02","updated_at":"2026-07-24T03:02:20Z","subjects":[],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/1903/3740","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Washington, Lawrence"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Steurer, Aliza Anne"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2006-09-12T05:38:26Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2006-09-12T05:38:26Z"]},{"key":"dc:date.issued","label":"Date","values":["2006-06-02"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/1903/3740"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf","application/postscript"]},{"key":"dc:title","label":"Title","values":["On the Galois Group of the 2-Class Field Towers of Some Imaginary Quadratic Fields"]}]}],"canonical_facts":{"dc:contributor.advisor":["Washington, Lawrence"],"dc:contributor.department":["Mathematics"],"dc:creator":["Steurer, Aliza Anne"],"dc:date.accessioned":["2006-09-12T05:38:26Z"],"dc:date.available":["2006-09-12T05:38:26Z"],"dc:date.issued":["2006-06-02"],"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."],"dc:format.mimetype":["application/pdf","application/postscript"],"dc:identifier.uri":["http://hdl.handle.net/1903/3740"],"dc:language.iso":["en_US"],"dc:title":["On the Galois Group of the 2-Class Field Towers of Some Imaginary Quadratic Fields"],"dc:type":["Dissertation"]},"updated_at":"2026-07-24T03:02:20Z"}