{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T09:38:04Z","timestamp":1777541884624,"version":"3.51.4"},"reference-count":23,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":4210,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Given any field<jats:italic>K<\/jats:italic>, there is a function field<jats:italic>F\/K<\/jats:italic>in one variable containing definable transcendental over<jats:italic>K<\/jats:italic>, i.e., elements in<jats:italic>F \/ K<\/jats:italic>first-order definable in the language of fields with parameters from<jats:italic>K<\/jats:italic>. Hence, the model-theoretic and the field-theoretic relative algebraic closure of<jats:italic>K<\/jats:italic>in<jats:italic>F do not<\/jats:italic>coincide. E.g., if<jats:italic>K<\/jats:italic>is finite, the model-theoretic algebraic closure of<jats:italic>K<\/jats:italic>in the rational function field<jats:italic>K(t)<\/jats:italic>is<jats:italic>K(t)<\/jats:italic>.<\/jats:p><jats:p>For the proof, diophantine \u2205-definability of<jats:italic>K<\/jats:italic>in<jats:italic>F<\/jats:italic>is established for any function field<jats:italic>F\/K<\/jats:italic>in one variable, provided<jats:italic>K<\/jats:italic>is large, or<jats:italic>K<\/jats:italic><jats:sup>\u00d7<\/jats:sup>\/(<jats:italic>K<\/jats:italic><jats:sup>\u00d7)<jats:italic>n<\/jats:italic><\/jats:sup>is finite for some integer<jats:italic>n<\/jats:italic>&gt; 1 coprime to<jats:italic>char K<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1190150142","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:14:26Z","timestamp":1197573266000},"page":"947-956","source":"Crossref","is-referenced-by-count":11,"title":["Defining transcendentals in function fields"],"prefix":"10.1017","volume":"67","author":[{"given":"Jochen","family":"Koenigsmann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120000935X_ref013","doi-asserted-by":"publisher","DOI":"10.2307\/2118581"},{"key":"S002248120000935X_ref021","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1920-8"},{"key":"S002248120000935X_ref019","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-69971-9"},{"key":"S002248120000935X_ref017","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1955-0070211-3"},{"key":"S002248120000935X_ref011","first-page":"354","article-title":"Enumerable sets are Diophantine","volume":"11","author":"Matiyasevich","year":"1970","journal-title":"Soviet Mathematics Doklady"},{"key":"S002248120000935X_ref010","doi-asserted-by":"publisher","DOI":"10.2969\/jmsj\/00310137"},{"key":"S002248120000935X_ref018","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1980-0583852-6"},{"key":"S002248120000935X_ref005","doi-asserted-by":"publisher","DOI":"10.1090\/surv\/006"},{"key":"S002248120000935X_ref015","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19640101803"},{"key":"S002248120000935X_ref012","doi-asserted-by":"publisher","DOI":"10.1007\/BF02402192"},{"key":"S002248120000935X_ref023","first-page":"235","volume-title":"Aus einem noch nicht ver\u00f6ffentlichten Briefe an Herrn P. Schwarz","volume":"II","author":"Weierstra\u00df"},{"key":"S002248120000935X_ref020","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1515\/crll.1938.179.5","article-title":"\u00dcber die Automorphismen eines algebraischen Funktionenk\u00f6rpers von Primzahlcharakteristik","volume":"179","author":"Schmid","year":"1938","journal-title":"Journal f\u00fcr die Reine und Angewandte Mathematik"},{"key":"S002248120000935X_ref016","doi-asserted-by":"publisher","DOI":"10.1007\/BF01899371"},{"key":"S002248120000935X_ref008","volume-title":"Algebraic Curves","author":"Fulton","year":"1969"},{"key":"S002248120000935X_ref007","first-page":"948","volume":"51","author":"Duret","year":"1986","journal-title":"Sur la th\u00e9orie \u00e9l\u00e9mentaire des corps de fonctions"},{"key":"S002248120000935X_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-68521-0_9"},{"key":"S002248120000935X_ref002","first-page":"87","volume":"59","author":"B\u00e9lair","year":"1994","journal-title":"Ind\u00e9cidabilit\u00e9 des corps de courbe r\u00e9elle"},{"key":"S002248120000935X_ref004","first-page":"107","article-title":"Definable sets over finite fields","volume":"427","author":"Chatzidakis","year":"1992","journal-title":"Journal f\u00fcr die Reine und Angewandte Mathematik"},{"key":"S002248120000935X_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/BF01443420"},{"key":"S002248120000935X_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1992-1095220-5"},{"key":"S002248120000935X_ref001","first-page":"783","volume":"56","author":"B\u00e9lair","year":"1991","journal-title":"D\u00e9finissabilit\u00e9 dans les corps de fonctions p-adiques"},{"key":"S002248120000935X_ref022","volume-title":"Algebraic function fields and codes","author":"Stichtenoth","year":"1993"},{"key":"S002248120000935X_ref014","first-page":"98","volume":"14","author":"Robinson","year":"1949","journal-title":"Definability and decision problems in arithmetic"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120000935X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,15]],"date-time":"2023-05-15T08:08:32Z","timestamp":1684138112000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120000935X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,9]]},"references-count":23,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2002,9]]}},"alternative-id":["S002248120000935X"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150142","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,9]]}}}