{"id":"https://openalex.org/W2886763533","doi":"https://doi.org/10.1080/00029890.2018.1470414","title":"What Happens When the Division Algorithm \u201cAlmost\u201d Works","display_name":"What Happens When the Division Algorithm \u201cAlmost\u201d Works","publication_year":2018,"publication_date":"2018-08-03","ids":{"openalex":"https://openalex.org/W2886763533","doi":"https://doi.org/10.1080/00029890.2018.1470414","mag":"2886763533"},"language":"en","primary_location":{"id":"doi:10.1080/00029890.2018.1470414","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2018.1470414","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5006699882","display_name":"Scott T. Chapman","orcid":"https://orcid.org/0000-0003-1205-6899"},"institutions":[{"id":"https://openalex.org/I191429286","display_name":"Sam Houston State University","ror":"https://ror.org/00yh3cz06","country_code":"US","type":"education","lineage":["https://openalex.org/I191429286"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Scott T. Chapman","raw_affiliation_strings":["Department of Mathematics and Statistics, Sam Houston State University, Box 2206, Huntsville, TX 77341"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, Sam Houston State University, Box 2206, Huntsville, TX 77341","institution_ids":["https://openalex.org/I191429286"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5006699882"],"corresponding_institution_ids":["https://openalex.org/I191429286"],"apc_list":null,"apc_paid":null,"fwci":0.965,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.72897402,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"125","issue":"7","first_page":"643","last_page":"647"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9984999895095825,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.996999979019165,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9846000075340271,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/division","display_name":"Division (mathematics)","score":0.8933665156364441},{"id":"https://openalex.org/keywords/division-algorithm","display_name":"Division algorithm","score":0.6789875030517578},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6132293343544006},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.5775307416915894},{"id":"https://openalex.org/keywords/key","display_name":"Key (lock)","score":0.5567726492881775},{"id":"https://openalex.org/keywords/principal","display_name":"Principal (computer security)","score":0.5397904515266418},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.5381271839141846},{"id":"https://openalex.org/keywords/generator","display_name":"Generator (circuit theory)","score":0.5247718691825867},{"id":"https://openalex.org/keywords/property","display_name":"Property (philosophy)","score":0.4953655004501343},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.48953232169151306},{"id":"https://openalex.org/keywords/integral-domain","display_name":"Integral domain","score":0.4553315341472626},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4395983815193176},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.43567079305648804},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.3784855604171753},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.2929683327674866},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.25074541568756104},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1039009690284729},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.06341767311096191}],"concepts":[{"id":"https://openalex.org/C60798267","wikidata":"https://www.wikidata.org/wiki/Q1226939","display_name":"Division (mathematics)","level":2,"score":0.8933665156364441},{"id":"https://openalex.org/C190715199","wikidata":"https://www.wikidata.org/wiki/Q3526714","display_name":"Division algorithm","level":3,"score":0.6789875030517578},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6132293343544006},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.5775307416915894},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.5567726492881775},{"id":"https://openalex.org/C144559511","wikidata":"https://www.wikidata.org/wiki/Q2986279","display_name":"Principal (computer security)","level":2,"score":0.5397904515266418},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.5381271839141846},{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.5247718691825867},{"id":"https://openalex.org/C189950617","wikidata":"https://www.wikidata.org/wiki/Q937228","display_name":"Property (philosophy)","level":2,"score":0.4953655004501343},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.48953232169151306},{"id":"https://openalex.org/C98486379","wikidata":"https://www.wikidata.org/wiki/Q628792","display_name":"Integral domain","level":3,"score":0.4553315341472626},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4395983815193176},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.43567079305648804},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.3784855604171753},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2929683327674866},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25074541568756104},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1039009690284729},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.06341767311096191},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1080/00029890.2018.1470414","is_oa":false,"landing_page_url":"https://doi.org/10.1080/00029890.2018.1470414","pdf_url":null,"source":{"id":"https://openalex.org/S156280010","display_name":"American Mathematical Monthly","issn_l":"0002-9890","issn":["0002-9890","1930-0972"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320547","host_organization_name":"Taylor & Francis","host_organization_lineage":["https://openalex.org/P4310320547"],"host_organization_lineage_names":["Taylor & Francis"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The American Mathematical Monthly","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":10,"referenced_works":["https://openalex.org/W382720801","https://openalex.org/W1510478584","https://openalex.org/W1591180289","https://openalex.org/W2042898177","https://openalex.org/W2109147142","https://openalex.org/W2963081530","https://openalex.org/W4213144751","https://openalex.org/W4237442181","https://openalex.org/W4248480617","https://openalex.org/W4253678486"],"related_works":["https://openalex.org/W2388178587","https://openalex.org/W2162724120","https://openalex.org/W1618607655","https://openalex.org/W2094996819","https://openalex.org/W2420477268","https://openalex.org/W4225492784","https://openalex.org/W1601585824","https://openalex.org/W243046325","https://openalex.org/W2153186231","https://openalex.org/W2980058475"],"abstract_inverted_index":{"Let":[0],"K":[1],"be":[2],"any":[3],"field.":[4],"The":[5],"division":[6,22,46],"algorithm":[7,23,47],"plays":[8],"a":[9,40],"key":[10],"role":[11],"in":[12],"studying":[13],"the":[14,21,27,45,74,78,85],"basic":[15],"algebraic":[16],"structure":[17],"of":[18,29,37,44,69,87],"K[X].":[19],"While":[20],"implies":[24],"that":[25,35,50],"all":[26],"ideals":[28,49],"K[X]":[30,38],"are":[31,54],"principal,":[32,53],"we":[33],"show":[34],"subrings":[36],"satisfying":[39],"slightly":[41],"weaker":[42],"version":[43],"produce":[48],"while":[51],"not":[52,77],"still":[55],"finitely":[56],"generated.":[57],"Our":[58],"construction":[59],"leads":[60],"to":[61,84],"an":[62,70],"example":[63],"for":[64],"each":[65],"positive":[66],"integer":[67],"n":[68,79],"integral":[71],"domain":[72],"with":[73],"n,":[75],"but":[76],"\u2212":[80],"1,":[81],"generator":[82],"property.Dedicated":[83],"Memory":[86],"Nick":[88],"Vaughan":[89]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
