{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T23:31:25Z","timestamp":1649201485279},"reference-count":3,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":12337,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1980,6]]},"abstract":"<jats:p>We consider the effect on decidability of adding, to the decidable theory of algebraically closed fields of characteristic zero, relation symbols for algebraic independence or function symbols for differentiation. Our results show that the corresponding theories are usually undeeidable.<\/jats:p><jats:p>Let <jats:italic>k<\/jats:italic> and <jats:italic>K<\/jats:italic> be algebraically closed fields of characteristic zero. Let <jats:italic>K<\/jats:italic> be an extension of <jats:italic>k<\/jats:italic> of transcendence degree <jats:italic>n<\/jats:italic> over <jats:italic>k<\/jats:italic>. Since <jats:italic>k<\/jats:italic> has characteristic 0, we may assume that the rational field, <jats:italic>Q<\/jats:italic>, is a subfield of <jats:italic>k<\/jats:italic>.<\/jats:p><jats:p>Let Ind<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> be the <jats:italic>n<\/jats:italic>-ary relation on <jats:italic>K<\/jats:italic> which holds for exactly those <jats:italic>n<\/jats:italic>-tuples from <jats:italic>K<\/jats:italic> which are algebraically independent over <jats:italic>k<\/jats:italic>.<\/jats:p><jats:p>Let <jats:italic>x<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>x<jats:sub>n<\/jats:sub><\/jats:italic> be a transcendence base for <jats:italic>K<\/jats:italic> over <jats:italic>k<\/jats:italic>. For <jats:italic>i<\/jats:italic> = 1, 2, \u2026, <jats:italic>n<\/jats:italic>, let <jats:italic>D<jats:sub>i<\/jats:sub><\/jats:italic>: <jats:italic>K<\/jats:italic> \u2192 <jats:italic>K<\/jats:italic> be the partial differentiation function with respect to <jats:italic>x<jats:sub>i<\/jats:sub><\/jats:italic> and this base.<\/jats:p><jats:p>Let <jats:italic>K<\/jats:italic><jats:sub arrange=\"stack\"><jats:italic>n<\/jats:italic><\/jats:sub><jats:sup arrange=\"stack\">Ind<\/jats:sup> = (<jats:italic>K<\/jats:italic>, +, \u00b7, Ind<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>), <jats:italic>n<\/jats:italic> \u2264 1 and let <jats:italic>K<\/jats:italic><jats:sub arrange=\"stack\"><jats:italic>n<\/jats:italic><\/jats:sub><jats:sup arrange=\"stack\">Diff<\/jats:sup> = (<jats:italic>K<\/jats:italic>, +, \u00b7, <jats:italic>D<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026, <jats:italic>D<jats:sub>n<\/jats:sub><\/jats:italic>), <jats:italic>n<\/jats:italic> \u2264 1 where <jats:italic>K<\/jats:italic> has transcendence degree <jats:italic>n<\/jats:italic> over <jats:italic>k<\/jats:italic>.<\/jats:p><jats:p>We show that the theories of these structures are independent of <jats:italic>k<\/jats:italic> when <jats:italic>k<\/jats:italic> has infinite transcendence degree over <jats:italic>Q<\/jats:italic>, that <jats:italic>K<\/jats:italic><jats:sub arrange=\"stack\"><jats:italic>n<\/jats:italic><\/jats:sub><jats:sup arrange=\"stack\">Diff<\/jats:sup> has undeeidable theory for <jats:italic>n<\/jats:italic> \u2264 1 and that <jats:italic>K<\/jats:italic><jats:sub arrange=\"stack\"><jats:italic>n<\/jats:italic><\/jats:sub><jats:sup arrange=\"stack\">Ind<\/jats:sup> has undeeidable theory for <jats:italic>n<\/jats:italic> \u2264 2. The theory of <jats:italic>K<\/jats:italic><jats:sub arrange=\"stack\">1<\/jats:sub><jats:sup arrange=\"stack\">Ind<\/jats:sup> is decidable.<\/jats:p>","DOI":"10.2307\/2273196","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:52:48Z","timestamp":1146937968000},"page":"359-362","source":"Crossref","is-referenced-by-count":1,"title":["Some theories associated with algebraically closed fields"],"prefix":"10.1017","volume":"45","author":[{"given":"Chris","family":"Ash","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John","family":"Rosenthal","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200046879_ref002","volume-title":"Methods of algebraic geometry","volume":"I","author":"Hodge","year":"1947"},{"key":"S0022481200046879_ref003","first-page":"98","volume":"14","author":"Robinson","year":"1949","journal-title":"Definability and decision problems in arithmetic"},{"key":"S0022481200046879_ref001","first-page":"5","volume-title":"Studies in model theory","author":"Barwise","year":"1973"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200046879","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,26]],"date-time":"2019-05-26T15:12:38Z","timestamp":1558883558000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200046879\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1980,6]]},"references-count":3,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1980,6]]}},"alternative-id":["S0022481200046879"],"URL":"https:\/\/doi.org\/10.2307\/2273196","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1980,6]]}}}