{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/72532"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/72532","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Steinitz Classes of Tamely Ramified Nonabelian Extensions of Algebraic Number Fields of Degree P(3)","abstract":"Let L/k be a finite extension of algebraic number fields of degree n with rings of integers ${\\cal D}\\sb{L}$ and ${\\cal D}\\sb{k}$. As an ${\\cal D}\\sb{k}$-module ${\\cal D}\\sb{L}$ is finitely generated and torsion free and thus can be written as a direct sum of $n - 1$ copies of ${\\cal D}\\sb{k}$ and a fractional ideal ${\\cal J}$ of k. The class $c\\ell({\\cal J})$ of ${\\cal J}$ in the ideal class group C(k) of k is the Steinitz class C(L,k) of ${\\cal D}\\sb{L}$. Now let G be a finite group of order n. As L varies over all normal extensions of k such that Gal(L/k) $\\simeq$ G, C(L,k) varies over a subset of C(k). These are the realizable classes of k with respect to G which we denote by R(k,G). If we consider only tamely ramified extensions then we denote this set by $R\\sb{t}(k,G)$. If G is an abelian group with exponent b, and k contains the multiplicative group of b-th roots of unity, then it is known that $R\\sb{t}(k,G)$ = $C(k)\\sp{m}$ (the subgroup of $C(k)$ consisting of m-th powers of elements of $C(k))$ where m is a positive rational integer which depends on the structure of G. We obtain a partial generalization of this result in the sense that if n = $p\\sp3$ where p is an odd prime then we can remove the restriction that G be an abelian group.","abstract_html":"Let L/k be a finite extension of algebraic number fields of degree n with rings of integers ${\\cal D}\\sb{L}$ and ${\\cal D}\\sb{k}$. As an ${\\cal D}\\sb{k}$-module ${\\cal D}\\sb{L}$ is finitely generated and torsion free and thus can be written as a direct sum of $n - 1$ copies of ${\\cal D}\\sb{k}$ and a fractional ideal ${\\cal J}$ of k. The class $c\\ell({\\cal J})$ of ${\\cal J}$ in the ideal class group C(k) of k is the Steinitz class C(L,k) of ${\\cal D}\\sb{L}$. Now let G be a finite group of order n. As L varies over all normal extensions of k such that Gal(L/k) $\\simeq$ G, C(L,k) varies over a subset of C(k). These are the realizable classes of k with respect to G which we denote by R(k,G). If we consider only tamely ramified extensions then we denote this set by $R\\sb{t}(k,G)$. If G is an abelian group with exponent b, and k contains the multiplicative group of b-th roots of unity, then it is known that $R\\sb{t}(k,G)$ = $C(k)\\sp{m}$ (the subgroup of $C(k)$ consisting of m-th powers of elements of $C(k))$ where m is a positive rational integer which depends on the structure of G. We obtain a partial generalization of this result in the sense that if n = $p\\sp3$ where p is an odd prime then we can remove the restriction that G be an abelian group.","abstract_has_math":true,"creators":["Carter, James Edgar"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["McCulloh, L.,"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-17T23:17:44Z","date_published":"2014-12-17T23:17:44Z","updated_at":"2026-07-22T22:26:07Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI9305482"],"render_values":[{"text":"(UMI)AAI9305482","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/72532","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["McCulloh, L.,"]},{"key":"dc:creator","label":"Author","values":["Carter, James Edgar"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-17T23:17:44Z","10000-01-01","1992"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/72532","(UMI)AAI9305482"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let L/k be a finite extension of algebraic number fields of degree n with rings of integers ${\\cal D}\\sb{L}$ and ${\\cal D}\\sb{k}$. As an ${\\cal D}\\sb{k}$-module ${\\cal D}\\sb{L}$ is finitely generated and torsion free and thus can be written as a direct sum of $n - 1$ copies of ${\\cal D}\\sb{k}$ and a fractional ideal ${\\cal J}$ of k. The class $c\\ell({\\cal J})$ of ${\\cal J}$ in the ideal class group C(k) of k is the Steinitz class C(L,k) of ${\\cal D}\\sb{L}$. Now let G be a finite group of order n. As L varies over all normal extensions of k such that Gal(L/k) $\\simeq$ G, C(L,k) varies over a subset of C(k). These are the realizable classes of k with respect to G which we denote by R(k,G). If we consider only tamely ramified extensions then we denote this set by $R\\sb{t}(k,G)$. If G is an abelian group with exponent b, and k contains the multiplicative group of b-th roots of unity, then it is known that $R\\sb{t}(k,G)$ = $C(k)\\sp{m}$ (the subgroup of $C(k)$ consisting of m-th powers of elements of $C(k))$ where m is a positive rational integer which depends on the structure of G. We obtain a partial generalization of this result in the sense that if n = $p\\sp3$ where p is an odd prime then we can remove the restriction that G be an abelian group.","Made available in DSpace on 2014-12-17T23:17:44Z (GMT). No. of bitstreams: 1 9305482.pdf: 1608438 bytes, checksum: dd15b174003a8cf0d33667c465d2bb7d (MD5) Previous issue date: 1992","Embargo set by: Seth Robbins for item 72700 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","54 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1992."]},{"key":"dc:title","label":"Title","values":["Steinitz Classes of Tamely Ramified Nonabelian Extensions of Algebraic Number Fields of Degree P(3)"]}]}],"canonical_facts":{"dc:contributor":["McCulloh, L.,"],"dc:creator":["Carter, James Edgar"],"dc:date":["2014-12-17T23:17:44Z","10000-01-01","1992"],"dc:description":["Let L/k be a finite extension of algebraic number fields of degree n with rings of integers ${\\cal D}\\sb{L}$ and ${\\cal D}\\sb{k}$. As an ${\\cal D}\\sb{k}$-module ${\\cal D}\\sb{L}$ is finitely generated and torsion free and thus can be written as a direct sum of $n - 1$ copies of ${\\cal D}\\sb{k}$ and a fractional ideal ${\\cal J}$ of k. The class $c\\ell({\\cal J})$ of ${\\cal J}$ in the ideal class group C(k) of k is the Steinitz class C(L,k) of ${\\cal D}\\sb{L}$. Now let G be a finite group of order n. As L varies over all normal extensions of k such that Gal(L/k) $\\simeq$ G, C(L,k) varies over a subset of C(k). These are the realizable classes of k with respect to G which we denote by R(k,G). If we consider only tamely ramified extensions then we denote this set by $R\\sb{t}(k,G)$. If G is an abelian group with exponent b, and k contains the multiplicative group of b-th roots of unity, then it is known that $R\\sb{t}(k,G)$ = $C(k)\\sp{m}$ (the subgroup of $C(k)$ consisting of m-th powers of elements of $C(k))$ where m is a positive rational integer which depends on the structure of G. We obtain a partial generalization of this result in the sense that if n = $p\\sp3$ where p is an odd prime then we can remove the restriction that G be an abelian group.","Made available in DSpace on 2014-12-17T23:17:44Z (GMT). No. of bitstreams: 1 9305482.pdf: 1608438 bytes, checksum: dd15b174003a8cf0d33667c465d2bb7d (MD5) Previous issue date: 1992","Embargo set by: Seth Robbins for item 72700 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","54 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1992."],"dc:identifier":["http://hdl.handle.net/2142/72532","(UMI)AAI9305482"],"dc:subject":["Mathematics"],"dc:title":["Steinitz Classes of Tamely Ramified Nonabelian Extensions of Algebraic Number Fields of Degree P(3)"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:07Z"}