{"id":"https://openalex.org/W2962905250","doi":"https://doi.org/10.2478/forma-2019-0010","title":"On Roots of Polynomials over <i>F</i> [ <i>X</i> ]/ \u3008 <i>p</i> \u3009","display_name":"On Roots of Polynomials over <i>F</i> [ <i>X</i> ]/ \u3008 <i>p</i> \u3009","publication_year":2019,"publication_date":"2019-07-01","ids":{"openalex":"https://openalex.org/W2962905250","doi":"https://doi.org/10.2478/forma-2019-0010","mag":"2962905250"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2019-0010","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2019-0010","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/27/2/article-p93.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/27/2/article-p93.pdf","any_repository_has_fulltext":false},"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":0.6064,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.7070434,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"27","issue":"2","first_page":"93","last_page":"100"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9994000196456909,"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"}},"topics":[{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9994000196456909,"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":0.9962000250816345,"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.9959999918937683,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/monomorphism","display_name":"Monomorphism","score":0.8158133029937744},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8097628951072693},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6380533576011658},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.602210283279419},{"id":"https://openalex.org/keywords/root-of-unity","display_name":"Root of unity","score":0.4853779077529907},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.4770343005657196},{"id":"https://openalex.org/keywords/functor","display_name":"Functor","score":0.4427539110183716},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.4387741684913635},{"id":"https://openalex.org/keywords/kronecker-delta","display_name":"Kronecker delta","score":0.42830905318260193},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3835761547088623},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.16050103306770325},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09825146198272705},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.09624448418617249}],"concepts":[{"id":"https://openalex.org/C100036529","wikidata":"https://www.wikidata.org/wiki/Q1945067","display_name":"Monomorphism","level":3,"score":0.8158133029937744},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8097628951072693},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6380533576011658},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.602210283279419},{"id":"https://openalex.org/C206343339","wikidata":"https://www.wikidata.org/wiki/Q756747","display_name":"Root of unity","level":3,"score":0.4853779077529907},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.4770343005657196},{"id":"https://openalex.org/C156772000","wikidata":"https://www.wikidata.org/wiki/Q864475","display_name":"Functor","level":2,"score":0.4427539110183716},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.4387741684913635},{"id":"https://openalex.org/C39482219","wikidata":"https://www.wikidata.org/wiki/Q192826","display_name":"Kronecker delta","level":2,"score":0.42830905318260193},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3835761547088623},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.16050103306770325},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09825146198272705},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.09624448418617249},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","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/C128107574","wikidata":"https://www.wikidata.org/wiki/Q182003","display_name":"Injective function","level":2,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2478/forma-2019-0010","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2019-0010","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/27/2/article-p93.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:cb49cfec655e4b3281d7be96f4bd9b36","is_oa":false,"landing_page_url":"https://doaj.org/article/cb49cfec655e4b3281d7be96f4bd9b36","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 2, Pp 93-100 (2019)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/8126","is_oa":false,"landing_page_url":"http://hdl.handle.net/11320/8126","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":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"}],"best_oa_location":{"id":"doi:10.2478/forma-2019-0010","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2019-0010","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/27/2/article-p93.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/W2962905250.pdf","grobid_xml":"https://content.openalex.org/works/W2962905250.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W1245804045","https://openalex.org/W2051422959","https://openalex.org/W2295728669","https://openalex.org/W2485566906","https://openalex.org/W2504155696","https://openalex.org/W2526342011","https://openalex.org/W2770430340","https://openalex.org/W3183392806"],"related_works":["https://openalex.org/W2963909210","https://openalex.org/W1592193675","https://openalex.org/W2092684517","https://openalex.org/W2953362199","https://openalex.org/W2061114238","https://openalex.org/W2089269623","https://openalex.org/W4394643047","https://openalex.org/W1973999653","https://openalex.org/W2021179827","https://openalex.org/W2033111719"],"abstract_inverted_index":{"Summary":[0],"This":[1],"is":[2,138,153,207,246,338,426,434,459],"the":[3,73,173,188,220,223,257,262,333,348,417,469,472,482,497],"first":[4,83,498],"part":[5,84,222,260,419,474,499],"of":[6,17,22,49,142,211,261,266,290,350,364,382,430,437,471,520],"a":[7,11,45,55,99,117,140,156,196,209,232,254,288,427,435,454,461,516,532],"four-article":[8],"series":[9],"containing":[10],"Mizar":[12],"[3],":[13],"[1],":[14],"[2]":[15],"formalization":[16,61],"Kronecker\u2019s":[18,63,312],"construction":[19,313,470],"about":[20],"roots":[21],"polynomials":[23],"in":[24,116,219,277,310,332,442,522],"field":[25,31,46,75,224,244,357,423,428,517],"extensions,":[26],"i.e.":[27],"that":[28,52,87,110,187,242,268,305,378,441],"for":[29,231,317,342,355,379,503],"every":[30,34,356],"F":[32,38,50,67,93,102,125,137,143,159,178,180,199,206,212,230,237,249,252,272,278,286,291,306,319,321,323,345,358,362,365,431,433,447,458,476,487,489,505,507,509,521,526],"and":[33,149,185,240,250,283,387,452],"polynomial":[35,90,157,190,455,462],"p":[36,53,91,132,152,165,171,193,456,524],"\u2208":[37,92,525],"[":[39,68,94,103,127,144,160,181,200,213,279,292,324,370,396,477,490,510,527],"X":[40,69,95,104,128,145,161,182,201,214,280,293,325,371,397,404,411,478,491,511,528],"]":[41,70,96,105,129,146,162,183,202,215,281,294,326,372,398,405,412,479,492,512,529],"\\F":[42,97,530],"there":[43],"exists":[44],"extension":[47,76,429,518],"E":[48,58,77,226,239,308,425,438,449,464,519],"such":[51],"has":[54,98,195,531],"root":[56,100,197],"over":[57,101,158,198,457,463],".":[59,107,167,204,439,465,494],"The":[60],"follows":[62],"classical":[64],"proof":[65,263],"using":[66],"/&lt;p&gt;":[71,106,147,184,203,216,282,295,480,493],"as":[72,134,253,287,329,450],"desired":[74],"[9],":[78],"[4],":[79],"[6].":[80],"In":[81,256,416],"this":[82,111,243,259,336,443,500],"we":[85,169,217,301,330,353,389,420,445],"show":[86,186,241,331],"an":[88,359],"irreducible":[89],"Note,":[108,440],"however,":[109],"statement":[112],"cannot":[113],"be":[114,315],"true":[115,341],"rigid":[118],"formal":[119],"sense:":[120],"We":[121,375,466],"do":[122,299],"not":[123,139,154,208,339],"have":[124,390,446],"\u2286":[126,448],"/":[130,163],"&lt;":[131,164],"&gt;":[133,166],"sets,":[135,451],"so":[136,300],"subfield":[141,210,289,436],",":[148,385],"hence":[150],"formally":[151],"even":[155],"Consequently,":[168],"translate":[170],"along":[172],"canonical":[174,483],"monomorphism":[175,234,484],"\u03d5":[176,191,235,485],":":[177,236,346,486],"\u2192":[179,238,488],"translated":[189],"(":[192,225],")":[194],"Because":[205],"construct":[218,354],"second":[221,473],"\\":[227,322],"\u03d5F":[228,276],")\u222a":[229],"given":[233],"both":[245],"isomorphic":[247,360],"to":[248,298,303,475],"includes":[251],"subfield.":[255],"literature":[258],"usually":[264],"consists":[265],"saying":[267],"\u201cone":[269],"can":[270,314],"identify":[271],"with":[273,320,366,481,496,506],"its":[274],"image":[275],"therefore":[284],"consider":[285],"\u201d.":[296],"Interestingly,":[297],"need":[302],"assume":[304],"\u2229":[307,368,393,402,409,508],"=\u2205,":[309],"particular":[311],"formalized":[316],"fields":[318,344,504],"=\u2205.":[327],"Surprisingly,":[328],"third":[334],"part,":[335],"condition":[337],"automatically":[340],"arbitray":[343],"With":[347],"exception":[349],"\ud835\udd51":[351,383,391,394],"2":[352],"copy":[361],"\u2032":[363],"F\u2032":[367,369],"\u2260":[373],"\u2205.":[374],"also":[376,460],"prove":[377],"Mizar\u2019s":[380],"representations":[381],"n":[384,392,395],"\ud835\udd48":[386,401],"\ud835\udd49":[388,408],"=":[399,406,413,513],"\u2205,":[400,414],"\ud835\udd48[":[403],"\u2205and":[407],"\ud835\udd49[":[410],"respectively.":[415],"fourth":[418],"finally":[421],"define":[422],"extensions:":[424],"i":[432],"case":[444],"thus":[453],"then":[467],"apply":[468],"Together":[495],"gives":[501],"-":[502,515],"\u2205":[514],"which":[523],"root.":[533]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2019-07-30T00:00:00"}
