{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,3,23]],"date-time":"2023-03-23T04:08:33Z","timestamp":1679544513331},"reference-count":13,"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":8228,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1991,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A chain-closed field is defined as a chainable field (i.e. a real field such that, for all <jats:italic>n<\/jats:italic> \u2208 N, \u03a3<jats:italic>K<\/jats:italic><jats:sup>2<jats:italic>n<\/jats:italic>+2<\/jats:sup> \u2260 \u03a3<jats:italic>K<\/jats:italic><jats:sup>2<jats:italic>n<\/jats:italic><\/jats:sup>) which does not admit any \u201cfaithful\u201d algebraic extension, and can also be seen as a field having a Henselian valuation \u03bd such that the residue field <jats:italic>K<\/jats:italic>\/\u03bd is real closed and the value group \u03bd<jats:italic>K<\/jats:italic> is odd divisible with \u2223\u03bd<jats:italic>K<\/jats:italic>\/2\u03bd<jats:italic>K<\/jats:italic>\u2223 = 2. If <jats:italic>K<\/jats:italic> admits only one such valuation, we show that <jats:italic>f<\/jats:italic> \u2208 <jats:italic>K<\/jats:italic>(<jats:italic>X<\/jats:italic>) is in \u03a3<jats:italic>K<\/jats:italic>(<jats:italic>X<\/jats:italic>)<jats:sup>2<jats:italic>n<\/jats:italic><\/jats:sup> for any real algebraic extension <jats:italic>L<\/jats:italic> of <jats:italic>K<\/jats:italic>,\u201c<jats:italic>f<\/jats:italic>(<jats:italic>L<\/jats:italic>) \u2286 \u03a3<jats:italic>L<\/jats:italic><jats:sup>2<jats:italic>n<\/jats:italic><\/jats:sup>\u201d holds. The conclusion is also true for <jats:italic>K<\/jats:italic> = <jats:bold>R<\/jats:bold>((<jats:italic>t<\/jats:italic>))(a chainable but not chain-closed field), and in the case <jats:italic>n<\/jats:italic> = 1 it holds for several variables and any real field <jats:italic>K<\/jats:italic>.<\/jats:p>","DOI":"10.2178\/jsl\/1183743733","type":"journal-article","created":{"date-parts":[[2009,8,4]],"date-time":"2009-08-04T20:52:04Z","timestamp":1249419124000},"page":"853-861","source":"Crossref","is-referenced-by-count":0,"title":["XVII<sup>\u00e8me<\/sup> probl\u00e8me de Hilbert sur les corps cha\u00eene-clos"],"prefix":"10.1017","volume":"56","author":[{"given":"Fran\u00e7oise Delon et Danielle","family":"Gondard","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200024026_ref012","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1960-0114855-0"},{"key":"S0022481200024026_ref011","volume-title":"Th\u00e9orie des valuations","author":"Ribenboim","year":"1964"},{"key":"S0022481200024026_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0083916"},{"key":"S0022481200024026_ref006","first-page":"921","volume":"53","author":"Dickmann","year":"1988","journal-title":"The model theory of chain closed fields"},{"key":"S0022481200024026_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1986-0835889-8"},{"key":"S0022481200024026_ref003","volume-title":"Quelques propri\u00e9t\u00e9s des corps valu\u00e9s en th\u00e9orie des mod\u00e8","author":"Delon","year":"1982"},{"key":"S0022481200024026_ref002","first-page":"77","article-title":"Rational points on algebraic varieties over a general real closed field: a model theoretic approach","volume":"357","author":"Becker","year":"1985","journal-title":"Journal f\u00fcr die Reine und Angewandte Mathematik"},{"key":"S0022481200024026_ref013","volume-title":"Model theory of ordered abelian groups","author":"Schmitt","year":"1982"},{"key":"S0022481200024026_ref001","volume-title":"Hereditarily-Pythagorean fields and orderings of higher types","volume":"29","author":"Becker","year":"1978"},{"key":"S0022481200024026_ref009","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/008\/653181"},{"key":"S0022481200024026_ref007","first-page":"463","article-title":"Th\u00e9orie du premier ordre des corps cha\u00eenables et des corps cha\u00eene-clos","volume":"304","author":"Gondard","year":"1987","journal-title":"Comptes Rendus des S\u00e9ances de l'Acad\u00e9mie des Sciences. S\u00e9rie I: Math\u00e9matique"},{"key":"S0022481200024026_ref010","first-page":"213","article-title":"The model theory of generalized real closed fields","volume":"323","author":"Jacob","year":"1981","journal-title":"Journal f\u00fcr die Reine und Angewandte Mathematik"},{"key":"S0022481200024026_ref005","first-page":"79","volume-title":"S\u00e9minaire sur les structures alg\u00e9briques ordonn\u00e9es","volume":"32","author":"Delon","year":"1990"}],"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200024026","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T10:34:31Z","timestamp":1679481271000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200024026\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1991,9]]},"references-count":13,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1991,9]]}},"alternative-id":["S0022481200024026"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1183743733","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1991,9]]}}}