{"id":"https://openalex.org/W3008758757","doi":"https://doi.org/10.2478/forma-2019-0022","title":"Field Extensions and Kronecker\u2019s Construction","display_name":"Field Extensions and Kronecker\u2019s Construction","publication_year":2019,"publication_date":"2019-10-01","ids":{"openalex":"https://openalex.org/W3008758757","doi":"https://doi.org/10.2478/forma-2019-0022","mag":"3008758757"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2019-0022","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2019-0022","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2019-0022","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://www.sciendo.com/pdf/10.2478/forma-2019-0022","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":true,"raw_author_name":"Christoph 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":["https://openalex.org/A5048465549"],"corresponding_institution_ids":["https://openalex.org/I40413290"],"apc_list":null,"apc_paid":null,"fwci":0.7254,"has_fulltext":true,"cited_by_count":10,"citation_normalized_percentile":{"value":0.79807855,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"27","issue":"3","first_page":"229","last_page":"235"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9980000257492065,"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"}},"topics":[{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.9980000257492065,"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"}},{"id":"https://openalex.org/T11424","display_name":"Security and Verification in Computing","score":0.991100013256073,"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"}},{"id":"https://openalex.org/T10142","display_name":"Formal Methods in Verification","score":0.98089998960495,"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/monomorphism","display_name":"Monomorphism","score":0.8167932033538818},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8167287111282349},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.6342521905899048},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6109153032302856},{"id":"https://openalex.org/keywords/kronecker-delta","display_name":"Kronecker delta","score":0.5462817549705505},{"id":"https://openalex.org/keywords/functor","display_name":"Functor","score":0.5072829127311707},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4588004946708679},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.4550330638885498},{"id":"https://openalex.org/keywords/root-of-unity","display_name":"Root of unity","score":0.44067469239234924},{"id":"https://openalex.org/keywords/field-extension","display_name":"Field extension","score":0.4297897219657898},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40816640853881836},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3228999376296997},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.18381786346435547},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10378643870353699},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08688056468963623},{"id":"https://openalex.org/keywords/injective-function","display_name":"Injective function","score":0.07771196961402893}],"concepts":[{"id":"https://openalex.org/C100036529","wikidata":"https://www.wikidata.org/wiki/Q1945067","display_name":"Monomorphism","level":3,"score":0.8167932033538818},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8167287111282349},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.6342521905899048},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6109153032302856},{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.5462817549705505},{"id":"https://openalex.org/C156772000","wikidata":"https://www.wikidata.org/wiki/Q864475","display_name":"Functor","level":2,"score":0.5072829127311707},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4588004946708679},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.4550330638885498},{"id":"https://openalex.org/C206343339","wikidata":"https://www.wikidata.org/wiki/Q756747","display_name":"Root of unity","level":3,"score":0.44067469239234924},{"id":"https://openalex.org/C103738662","wikidata":"https://www.wikidata.org/wiki/Q577835","display_name":"Field extension","level":3,"score":0.4297897219657898},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40816640853881836},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3228999376296997},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.18381786346435547},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10378643870353699},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08688056468963623},{"id":"https://openalex.org/C128107574","wikidata":"https://www.wikidata.org/wiki/Q182003","display_name":"Injective function","level":2,"score":0.07771196961402893},{"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/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2478/forma-2019-0022","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2019-0022","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2019-0022","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:d7c8673a7af443e5804292ae24345015","is_oa":false,"landing_page_url":"https://doaj.org/article/d7c8673a7af443e5804292ae24345015","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 27, Iss 3, Pp 229-235 (2019)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/9013","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/9013","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":"","raw_type":"Article"}],"best_oa_location":{"id":"doi:10.2478/forma-2019-0022","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2019-0022","pdf_url":"https://www.sciendo.com/pdf/10.2478/forma-2019-0022","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/W3008758757.pdf","grobid_xml":"https://content.openalex.org/works/W3008758757.grobid-xml"},"referenced_works_count":5,"referenced_works":["https://openalex.org/W2184838578","https://openalex.org/W2485566906","https://openalex.org/W2526342011","https://openalex.org/W2770430340","https://openalex.org/W3183392806"],"related_works":["https://openalex.org/W2963909210","https://openalex.org/W3114459035","https://openalex.org/W1592193675","https://openalex.org/W2092684517","https://openalex.org/W1973999653","https://openalex.org/W2021179827","https://openalex.org/W3008758757","https://openalex.org/W3008655298","https://openalex.org/W4313484603","https://openalex.org/W4313308154"],"abstract_inverted_index":{"Summary":[0],"This":[1],"is":[2,139,154,208,247,341,429,437,462],"the":[3,73,82,174,189,221,224,258,263,336,351,472,475,485,500],"fourth":[4,421],"part":[5,84,223,261,422,477,502],"of":[6,17,22,49,143,212,262,267,291,353,366,384,433,440,474,523],"a":[7,11,45,55,99,117,141,157,197,210,233,255,289,430,438,457,464,519],"four-article":[8],"series":[9],"containing":[10],"Mizar":[12],"[3],":[13],"[2],":[14],"[1]":[15],"formalization":[16,61],"Kronecker\u2019s":[18,63,314],"construction":[19,315,473],"about":[20],"roots":[21],"polynomials":[23],"in":[24,116,220,278,312,335,445,525],"field":[25,31,46,75,225,245,360,426,431,520],"extensions,":[26],"i.e.":[27],"that":[28,52,87,110,188,243,269,306,380,444],"for":[29,232,319,345,358,381,506],"every":[30,34,359],"F":[32,38,50,67,93,102,125,127,138,144,160,179,181,200,207,213,231,238,250,253,273,279,287,292,307,321,323,325,348,361,367,434,436,450,461,479,490,492,508,510,512,524,529],"and":[33,150,186,241,251,284,389,410,455],"polynomial":[35,90,158,191,458,465],"p":[36,53,91,133,153,166,172,194,459,527],"\u2208":[37,92,528],"[":[39,68,94,103,128,145,161,182,201,214,280,293,326,372,398,480,493,513,530],"X":[40,69,95,104,129,146,162,183,202,215,281,294,327,373,399,406,414,481,494,514,531],"]":[41,70,96,105,130,147,163,184,203,216,282,295,328,374,400,407,415,482,495,515,532],"\\F":[42,97],"there":[43],"exists":[44],"extension":[47,76,432,521],"E":[48,58,77,227,240,309,428,441,452,467,522],"such":[51],"has":[54,98,196],"root":[56,100,198],"over":[57,101,159,199,460,466],".":[59,107,168,205,442,468,497],"The":[60],"follows":[62],"classical":[64],"proof":[65,264],"using":[66],"/&lt;p&gt;":[71,106,148,185,204,217,283,296,483,496],"as":[72,135,254,288,332,453],"desired":[74],"[6],":[78],"[4],":[79],"[5].":[80],"In":[81,257,419],"first":[83,501],"we":[85,170,218,302,333,356,391,423,448],"show":[86,187,242,334],"an":[88,362],"irreducible":[89],"Note,":[108,443],"however,":[109],"this":[111,244,260,339,420,446,503],"statement":[112],"cannot":[113],"be":[114,317],"true":[115,344],"rigid":[118],"formal":[119],"sense:":[120],"We":[121,377,469],"do":[122,300],"not":[123,140,155,209,342],"have":[124,392,449],"\u2286":[126,451],"/":[131,164],"&lt;":[132,165],"&gt;":[134,167],"sets,":[136,454],"so":[137,301],"subfield":[142,211,290,439],",":[149,387],"hence":[151],"formally":[152],"even":[156],"Consequently,":[169],"translate":[171],"along":[173],"canonical":[175,486],"monomorphism":[176,235,487],"\u03d5":[177,192,236,488],":":[178,237,349,489],"\u2192":[180,239,491],"translated":[190],"(":[193,226],")":[195],"Because":[206],"construct":[219,357],"second":[222,476],"\\":[228],"\u03d5F":[229,277],")\u222a":[230],"given":[234],"both":[246],"isomorphic":[248,363],"to":[249,299,304,478],"includes":[252],"subfield.":[256],"literature":[259],"usually":[265],"consists":[266],"saying":[268],"\u201cone":[270],"can":[271,316],"identify":[272],"with":[274,322,368,484,499,509],"its":[275],"image":[276],"therefore":[285],"consider":[286],"\u201d.":[297],"Interestingly,":[298],"need":[303],"assume":[305],"\u2229":[308,324,370,395,404,412,511],"=":[310,329,401,408,416,516],"\u2205,":[311,402,417],"particular":[313],"formalized":[318],"fields":[320,347,507],"\u2205.":[330,376],"Surprisingly,":[331],"third":[337],"part,":[338],"condition":[340],"automatically":[343],"arbitrary":[346],"With":[350],"exception":[352],"\u2124":[354,385,393,396],"2":[355],"copy":[364],"F\u2032":[365,369,371],"\u2260":[375],"also":[378,463],"prove":[379],"Mizar\u2019s":[382],"representations":[383],"n":[386,394,397],"\u211a":[388,403],"\u211d":[390,411],"\u211a[":[405],"\u2205":[409,517],"\u211d[":[413],"respectively.":[418],"finally":[424],"define":[425],"extensions:":[427],"iff":[435],"case":[447],"thus":[456],"then":[470],"apply":[471],"Together":[498],"gives":[504],"\u2013":[505,518],"which":[526]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2020-03-06T00:00:00"}
