{"id":"https://openalex.org/W2794816057","doi":"https://doi.org/10.1515/forma-2017-0024","title":"Formally Real Fields","display_name":"Formally Real Fields","publication_year":2017,"publication_date":"2017-12-20","ids":{"openalex":"https://openalex.org/W2794816057","doi":"https://doi.org/10.1515/forma-2017-0024","mag":"2794816057"},"language":"en","primary_location":{"id":"doi:10.1515/forma-2017-0024","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2017-0024","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/25/4/article-p249.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/25/4/article-p249.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"]}],"countries":["PL"],"is_corresponding":true,"raw_author_name":"Christoph Schwarzweller","raw_affiliation_strings":["Institute of Informatics Faculty of Mathematics, Physics and Informatics University of Gdansk Wita Stwosza 57, 80-308 Gdansk , Poland"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Informatics Faculty of Mathematics, Physics and Informatics University of Gdansk Wita Stwosza 57, 80-308 Gdansk , 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":0.0,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.33593402,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"25","issue":"4","first_page":"249","last_page":"259"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11703","display_name":"Rings, Modules, and Algebras","score":0.9973000288009644,"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.9973000288009644,"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.9957000017166138,"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/T12396","display_name":"Advanced Differential Equations and Dynamical Systems","score":0.9957000017166138,"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/mathematics","display_name":"Mathematics","score":0.8238828778266907},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.5936237573623657},{"id":"https://openalex.org/keywords/square-root","display_name":"Square root","score":0.5858783721923828},{"id":"https://openalex.org/keywords/ordered-field","display_name":"Ordered field","score":0.520093560218811},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.5089738368988037},{"id":"https://openalex.org/keywords/field","display_name":"Field (mathematics)","score":0.4781128764152527},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.44711771607398987},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.44207826256752014},{"id":"https://openalex.org/keywords/real-number","display_name":"Real number","score":0.43145084381103516},{"id":"https://openalex.org/keywords/algebraic-number-field","display_name":"Algebraic number field","score":0.4231053292751312},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.4177634119987488},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.3689102530479431},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.13587775826454163},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.08075681328773499}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8238828778266907},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.5936237573623657},{"id":"https://openalex.org/C11577676","wikidata":"https://www.wikidata.org/wiki/Q134237","display_name":"Square root","level":2,"score":0.5858783721923828},{"id":"https://openalex.org/C13851312","wikidata":"https://www.wikidata.org/wiki/Q909412","display_name":"Ordered field","level":2,"score":0.520093560218811},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.5089738368988037},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.4781128764152527},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.44711771607398987},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.44207826256752014},{"id":"https://openalex.org/C94020503","wikidata":"https://www.wikidata.org/wiki/Q12916","display_name":"Real number","level":2,"score":0.43145084381103516},{"id":"https://openalex.org/C12657307","wikidata":"https://www.wikidata.org/wiki/Q616608","display_name":"Algebraic number field","level":2,"score":0.4231053292751312},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.4177634119987488},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.3689102530479431},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.13587775826454163},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.08075681328773499}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1515/forma-2017-0024","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2017-0024","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/25/4/article-p249.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:ac6a021198694f6b8d63e3cf79e4a79c","is_oa":false,"landing_page_url":"https://doaj.org/article/ac6a021198694f6b8d63e3cf79e4a79c","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 25, Iss 4, Pp 249-259 (2017)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/6550","is_oa":true,"landing_page_url":"http://hdl.handle.net/11320/6550","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-2017-0024","is_oa":true,"landing_page_url":"https://doi.org/10.1515/forma-2017-0024","pdf_url":"https://content.sciendo.com/downloadpdf/journals/forma/25/4/article-p249.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":[{"id":"https://metadata.un.org/sdg/10","score":0.4399999976158142,"display_name":"Reduced inequalities"}],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2794816057.pdf","grobid_xml":"https://content.openalex.org/works/W2794816057.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W1553126454","https://openalex.org/W2057691643","https://openalex.org/W2184838578","https://openalex.org/W2326165709","https://openalex.org/W2526342011","https://openalex.org/W2614270863","https://openalex.org/W2777606786","https://openalex.org/W2777805251"],"related_works":["https://openalex.org/W2949318403","https://openalex.org/W2129886174","https://openalex.org/W2750949216","https://openalex.org/W2036109498","https://openalex.org/W2110665728","https://openalex.org/W2808623850","https://openalex.org/W4249661706","https://openalex.org/W2152143338","https://openalex.org/W3210602613","https://openalex.org/W1628504494"],"abstract_inverted_index":{"Summary":[0],"We":[1],"extend":[2],"the":[3,45,57,61,68],"algebraic":[4],"theory":[5],"of":[6,39,53,60],"ordered":[7,32],"fields":[8,33],"[7,":[9],"6]":[10],"in":[11,43],"Mizar":[12],"[1,":[13],"2,":[14],"3]:":[15],"we":[16,63],"show":[17],"that":[18,28],"every":[19],"preordering":[20],"can":[21],"be":[22],"extended":[23],"into":[24],"an":[25],"ordering,":[26],"i.e.":[27],"formally":[29,40],"real":[30,41],"and":[31,49,67],"coincide.We":[34],"further":[35],"prove":[36],"some":[37],"characterizations":[38],"fields,":[42],"particular":[44],"one":[46],"by":[47],"Artin":[48],"Schreier":[50],"using":[51],"sums":[52],"squares":[54],"[4].":[55],"In":[56],"second":[58],"part":[59],"article":[62],"define":[64],"absolute":[65],"values":[66],"square":[69],"root":[70],"function":[71],"[5].":[72]},"counts_by_year":[{"year":2021,"cited_by_count":1},{"year":2016,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}