{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/39950"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/39950","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Irreducible elements in algebraic number fields","abstract":"This dissertation is a study of two basic questions involving irreducible elements in algebraic number fields. The first question is: Given an algebraic integer β in a field with class number greater than two, how many different lengths of factorizations into irreducibles exist? The distribution into ideal classes of the prime ideals whose product is the principal ideal (β) determines the possible length of the factorizations into irreducibles. Chapter 2 gives precise answers when the field has class number 3 or 4, as well as when the class group is an elementary 2-group of order 8. The second question is: In a normal extension, when are there rational primes which split completely and remain irreducible? Chapter 3 focusses on the bicyclic bi-quadratic fields. The imaginary bicyclic biquadratic fields which contain such primes are completely determined.","abstract_html":"This dissertation is a study of two basic questions involving irreducible elements in algebraic number fields. The first question is: Given an algebraic integer β in a field with class number greater than two, how many different lengths of factorizations into irreducibles exist? The distribution into ideal classes of the prime ideals whose product is the principal ideal (β) determines the possible length of the factorizations into irreducibles. Chapter 2 gives precise answers when the field has class number 3 or 4, as well as when the class group is an elementary 2-group of order 8. The second question is: In a normal extension, when are there rational primes which split completely and remain irreducible? Chapter 3 focusses on the bicyclic bi-quadratic fields. The imaginary bicyclic biquadratic fields which contain such primes are completely determined.","abstract_has_math":false,"creators":["McCoy, Daisy Cox"],"institution":"Virginia Tech","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Parry, Charles J."],"committee_members":["Brown, E.","Farkas, Diana","Fletcher, Peter","Wheeler, Robert"],"year":1990,"date_issued":"1990","date_published":"1990","updated_at":"2026-07-22T22:19:16Z","subjects":[],"languages":["en"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-10192005-113254"],"render_values":[{"text":"etd-10192005-113254","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/39950","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Parry, Charles J."]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Brown, E.","Farkas, Diana","Fletcher, Peter","Wheeler, Robert"]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["McCoy, Daisy Cox"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2014-03-14T21:21:28Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2014-03-14T21:21:28Z","2005-10-19"]},{"key":"dc:date.issued","label":"Date","values":["1990"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["etd-10192005-113254"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/39950"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This dissertation is a study of two basic questions involving irreducible elements in algebraic number fields. The first question is: Given an algebraic integer β in a field with class number greater than two, how many different lengths of factorizations into irreducibles exist? The distribution into ideal classes of the prime ideals whose product is the principal ideal (β) determines the possible length of the factorizations into irreducibles. Chapter 2 gives precise answers when the field has class number 3 or 4, as well as when the class group is an elementary 2-group of order 8. The second question is: In a normal extension, when are there rational primes which split completely and remain irreducible? Chapter 3 focusses on the bicyclic bi-quadratic fields. The imaginary bicyclic biquadratic fields which contain such primes are completely determined."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["BTD"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Irreducible elements in algebraic number fields"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Parry, Charles J."],"dc:contributor.committeemember":["Brown, E.","Farkas, Diana","Fletcher, Peter","Wheeler, Robert"],"dc:contributor.department":["Mathematics"],"dc:creator":["McCoy, Daisy Cox"],"dc:date.accessioned":["2014-03-14T21:21:28Z"],"dc:date.available":["2014-03-14T21:21:28Z","2005-10-19"],"dc:date.issued":["1990"],"dc:description.abstract":["This dissertation is a study of two basic questions involving irreducible elements in algebraic number fields. The first question is: Given an algebraic integer β in a field with class number greater than two, how many different lengths of factorizations into irreducibles exist? The distribution into ideal classes of the prime ideals whose product is the principal ideal (β) determines the possible length of the factorizations into irreducibles. Chapter 2 gives precise answers when the field has class number 3 or 4, as well as when the class group is an elementary 2-group of order 8. The second question is: In a normal extension, when are there rational primes which split completely and remain irreducible? Chapter 3 focusses on the bicyclic bi-quadratic fields. The imaginary bicyclic biquadratic fields which contain such primes are completely determined."],"dc:description.degree":["Ph. D."],"dc:format.medium":["BTD"],"dc:format.mimetype":["application/pdf"],"dc:identifier.other":["etd-10192005-113254"],"dc:identifier.uri":["http://hdl.handle.net/10919/39950"],"dc:language.iso":["en"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Irreducible elements in algebraic number fields"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:16Z"}