{"id":"https://openalex.org/W2588251761","doi":"https://doi.org/10.1515/forma-2016-0019","title":"Some Algebraic Properties of Polynomial Rings","display_name":"Some Algebraic Properties of Polynomial Rings","publication_year":2016,"publication_date":"2016-09-01","ids":{"openalex":"https://openalex.org/W2588251761","doi":"https://doi.org/10.1515/forma-2016-0019","mag":"2588251761"},"language":"en","primary_location":{"id":"doi:10.1515/forma-2016-0019","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2016-0019","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/24/3/article-p227.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://content.sciendo.com/downloadpdf/journals/forma/24/3/article-p227.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5048465549","display_name":"Christoph Schwarzweller","orcid":"https://orcid.org/0000-0001-9587-8737"},"institutions":[{"id":"https://openalex.org/I40413290","display_name":"University of Gda\u0144sk","ror":"https://ror.org/011dv8m48","country_code":"PL","type":"education","lineage":["https://openalex.org/I40413290"]},{"id":"https://openalex.org/I4210087266","display_name":"Institute of Computer Science","ror":"https://ror.org/003fvp964","country_code":"PL","type":"facility","lineage":["https://openalex.org/I4210087266","https://openalex.org/I99542240"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Christoph Schwarzweller","raw_affiliation_strings":["Institute of Computer Science University of Gdansk, Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Computer Science University of Gdansk, Poland","institution_ids":["https://openalex.org/I40413290","https://openalex.org/I4210087266"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5090121372","display_name":"Artur Korni\u0142owicz","orcid":"https://orcid.org/0000-0002-4565-9082"},"institutions":[{"id":"https://openalex.org/I245364917","display_name":"University of Bia\u0142ystok","ror":"https://ror.org/01qaqcf60","country_code":"PL","type":"education","lineage":["https://openalex.org/I245364917"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Artur Korni\u0142owicz","raw_affiliation_strings":["Institute of Informatics University of Bia\u0142ystok, Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Informatics University of Bia\u0142ystok, Poland","institution_ids":["https://openalex.org/I245364917"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5052622605","display_name":"Agnieszka Rowi\u0144ska-Schwarzweller","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Agnieszka Rowinska-Schwarzweller","raw_affiliation_strings":["Sopot Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Sopot Poland","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.5983,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.82855573,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"24","issue":"3","first_page":"227","last_page":"237"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9997000098228455,"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"}},"topics":[{"id":"https://openalex.org/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9997000098228455,"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"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9977999925613403,"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/T11130","display_name":"Coding theory and cryptography","score":0.995199978351593,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/monic-polynomial","display_name":"Monic polynomial","score":0.9284586906433105},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8698509931564331},{"id":"https://openalex.org/keywords/irreducible-polynomial","display_name":"Irreducible polynomial","score":0.7184540033340454},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.5570677518844604},{"id":"https://openalex.org/keywords/embedding","display_name":"Embedding","score":0.5183064341545105},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5089526176452637},{"id":"https://openalex.org/keywords/reciprocal-polynomial","display_name":"Reciprocal polynomial","score":0.504407525062561},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.4855198860168457},{"id":"https://openalex.org/keywords/factorization-of-polynomials","display_name":"Factorization of polynomials","score":0.47675830125808716},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.47448280453681946},{"id":"https://openalex.org/keywords/unit","display_name":"Unit (ring theory)","score":0.4593273997306824},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.43632179498672485},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.430157870054245},{"id":"https://openalex.org/keywords/minimal-polynomial","display_name":"Minimal polynomial (linear algebra)","score":0.42941814661026},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4196653962135315},{"id":"https://openalex.org/keywords/matrix-polynomial","display_name":"Matrix polynomial","score":0.2555731236934662},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.0975295901298523},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.051412343978881836}],"concepts":[{"id":"https://openalex.org/C163635466","wikidata":"https://www.wikidata.org/wiki/Q3099696","display_name":"Monic polynomial","level":3,"score":0.9284586906433105},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8698509931564331},{"id":"https://openalex.org/C165030954","wikidata":"https://www.wikidata.org/wiki/Q1476663","display_name":"Irreducible polynomial","level":4,"score":0.7184540033340454},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.5570677518844604},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.5183064341545105},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5089526176452637},{"id":"https://openalex.org/C97395012","wikidata":"https://www.wikidata.org/wiki/Q1422356","display_name":"Reciprocal polynomial","level":4,"score":0.504407525062561},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.4855198860168457},{"id":"https://openalex.org/C108741685","wikidata":"https://www.wikidata.org/wiki/Q392477","display_name":"Factorization of polynomials","level":4,"score":0.47675830125808716},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.47448280453681946},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.4593273997306824},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.43632179498672485},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.430157870054245},{"id":"https://openalex.org/C30597210","wikidata":"https://www.wikidata.org/wiki/Q1163608","display_name":"Minimal polynomial (linear algebra)","level":5,"score":0.42941814661026},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4196653962135315},{"id":"https://openalex.org/C101044782","wikidata":"https://www.wikidata.org/wiki/Q6787887","display_name":"Matrix polynomial","level":3,"score":0.2555731236934662},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0975295901298523},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.051412343978881836},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C145420912","wikidata":"https://www.wikidata.org/wiki/Q853077","display_name":"Mathematics education","level":1,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1515/forma-2016-0019","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2016-0019","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/24/3/article-p227.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:90372eb1409d4a7e9ccabbc43d2a1037","is_oa":false,"landing_page_url":"https://doaj.org/article/90372eb1409d4a7e9ccabbc43d2a1037","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Formalized Mathematics, Vol 24, Iss 3, Pp 227-237 (2016)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/5557","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/5557","pdf_url":null,"source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"}],"best_oa_location":{"id":"doi:10.1515/forma-2016-0019","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2016-0019","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/24/3/article-p227.pdf","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2588251761.pdf","grobid_xml":"https://content.openalex.org/works/W2588251761.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W1586230637","https://openalex.org/W2295728669","https://openalex.org/W4234679232","https://openalex.org/W4254183908"],"related_works":["https://openalex.org/W1576380577","https://openalex.org/W4249593543","https://openalex.org/W2068749660","https://openalex.org/W2937328185","https://openalex.org/W2003536029","https://openalex.org/W1985839497","https://openalex.org/W2049824936","https://openalex.org/W2054200117","https://openalex.org/W3090081616","https://openalex.org/W2588251761"],"abstract_inverted_index":{"Abstract":[0],"In":[1],"this":[2],"article":[3],"we":[4,26],"extend":[5],"the":[6,28,58,64,72],"algebraic":[7],"theory":[8],"of":[9,31,66,74],"polynomial":[10,46],"rings,":[11],"formalized":[12],"in":[13],"Mizar":[14],"[1],":[15],"based":[16],"on":[17],"[2],":[18],"[3].":[19],"After":[20],"introducing":[21],"constant":[22],"and":[23,35,40,48,54],"monic":[24],"polynomials":[25,56,67],"present":[27],"canonical":[29],"embedding":[30],"R":[32],"into":[33],"R[X]":[34],"deal":[36],"with":[37,68],"both":[38],"unit":[39],"irreducible":[41,55],"elements.":[42],"We":[43],"also":[44],"define":[45],"GCDs":[47],"show":[49],"that":[50],"for":[51],"fields":[52],"F":[53],"p":[57],"field":[59,65],"F[X]/&lt;p&gt;":[60],"is":[61],"isomorphic":[62],"to":[63],"degree":[69],"smaller":[70],"than":[71],"one":[73],"p.":[75]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2017-02-24T00:00:00"}
