{"id":"https://openalex.org/W4390622026","doi":"https://doi.org/10.2478/forma-2023-0023","title":"Simple Extensions","display_name":"Simple Extensions","publication_year":2023,"publication_date":"2023-09-01","ids":{"openalex":"https://openalex.org/W4390622026","doi":"https://doi.org/10.2478/forma-2023-0023"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2023-0023","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2023-0023","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2023-0023","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"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://sciendo.com/pdf/10.2478/forma-2023-0023","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"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Christoph Schwarzweller","raw_affiliation_strings":["Institute of Informatics , University of Gda\u0144sk , Poland"],"raw_orcid":"https://orcid.org/0000-0001-9587-8737","affiliations":[{"raw_affiliation_string":"Institute of Informatics , University of Gda\u0144sk , Poland","institution_ids":["https://openalex.org/I40413290"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5052622605","display_name":"Agnieszka Rowi\u0144ska-Schwarzweller","orcid":null},"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"]}],"countries":["PL"],"is_corresponding":false,"raw_author_name":"Agnieszka Rowi\u0144ska-Schwarzweller","raw_affiliation_strings":["Institute of Informatics , University of Gda\u0144sk , Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Informatics , University of Gda\u0144sk , Poland","institution_ids":["https://openalex.org/I40413290"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I40413290"],"apc_list":null,"apc_paid":null,"fwci":2.5976,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.90516302,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":91,"max":98},"biblio":{"volume":"31","issue":"1","first_page":"287","last_page":"298"},"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":0.9966999888420105,"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":0.9966999888420105,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.9843000173568726,"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":0.9821000099182129,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.8763995170593262},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8648906946182251},{"id":"https://openalex.org/keywords/divisibility-rule","display_name":"Divisibility rule","score":0.7646884918212891},{"id":"https://openalex.org/keywords/extension","display_name":"Extension (predicate logic)","score":0.6835065484046936},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.6475790739059448},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.599511444568634},{"id":"https://openalex.org/keywords/field-extension","display_name":"Field extension","score":0.5394253730773926},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.4731767475605011},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4629126191139221},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.41492384672164917},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08485645055770874}],"concepts":[{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.8763995170593262},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8648906946182251},{"id":"https://openalex.org/C9991821","wikidata":"https://www.wikidata.org/wiki/Q858520","display_name":"Divisibility rule","level":2,"score":0.7646884918212891},{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.6835065484046936},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.6475790739059448},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.599511444568634},{"id":"https://openalex.org/C103738662","wikidata":"https://www.wikidata.org/wiki/Q577835","display_name":"Field extension","level":3,"score":0.5394253730773926},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4731767475605011},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4629126191139221},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.41492384672164917},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08485645055770874},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.0},{"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/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2478/forma-2023-0023","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2023-0023","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2023-0023","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:repozytorium.uwb.edu.pl:11320/15862","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/15862","pdf_url":"https://repozytorium.uwb.edu.pl/jspui/bitstream/11320/15862/1/Simple-Extensions.pdf","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":"","raw_type":"Article"}],"best_oa_location":{"id":"doi:10.2478/forma-2023-0023","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2023-0023","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2023-0023","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":false},"content_urls":{"pdf":"https://content.openalex.org/works/W4390622026.pdf"},"referenced_works_count":11,"referenced_works":["https://openalex.org/W1625130818","https://openalex.org/W2480281325","https://openalex.org/W2485566906","https://openalex.org/W2526342011","https://openalex.org/W2588251761","https://openalex.org/W2777606786","https://openalex.org/W3124299171","https://openalex.org/W3159418482","https://openalex.org/W3197142639","https://openalex.org/W4206350999","https://openalex.org/W4388142980"],"related_works":["https://openalex.org/W245125619","https://openalex.org/W2068736327","https://openalex.org/W2081314644","https://openalex.org/W4246881104","https://openalex.org/W1835326893","https://openalex.org/W4295094445","https://openalex.org/W2347268433","https://openalex.org/W2147352225","https://openalex.org/W4287662791","https://openalex.org/W1996139403"],"abstract_inverted_index":{"Summary":[0],"In":[1],"this":[2,97],"article":[3],"we":[4,52,80,98],"continue":[5],"the":[6],"formalization":[7],"of":[8,20,34,59,85,119,126],"field":[9,87],"theory":[10],"in":[11],"Mizar.":[12],"We":[13],"introduce":[14],"simple":[15,23,62,94],"extensions:":[16],"an":[17],"extension":[18,57],"E":[19,25,35,39,49,58,75],"F":[21,29,41,60,77,88,106],"is":[22,26,38,61],"if":[24,63,66],"generated":[27],"over":[28,105],"by":[30],"a":[31,43,47,55,86],"single":[32,108],"element":[33],",":[36],"that":[37,54,82,102],"=":[40],"(":[42],")":[44],"for":[45],"some":[46],"\u2208":[48],".":[50],"First,":[51],"prove":[53],"finite":[56,83],"and":[64,76,117,123],"only":[65,69],"there":[67],"are":[68,92],"finitely":[70],"many":[71],"intermediate":[72],"fields":[73],"between":[74],"[7].":[78],"Second,":[79],"show":[81],"extensions":[84],"with":[89],"characteristic":[90],"0":[91],"always":[93],"[1].":[95],"For":[96],"had":[99],"to":[100],"prove,":[101],"irreducible":[103],"polynomials":[104,120,127],"have":[107],"roots":[109],"only,":[110],"which":[111],"required":[112],"extending":[113],"results":[114],"on":[115],"divisibility":[116],"gcds":[118],"[14],":[121],"[13]":[122],"formal":[124],"derivation":[125],"[15].":[128]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2}],"updated_date":"2026-06-26T08:34:08.712188","created_date":"2024-01-07T00:00:00"}
