{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T08:58:03Z","timestamp":1760345883924},"reference-count":22,"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":6859,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1995,6]]},"abstract":"<jats:p>On sait que par le choix d\u2019 un langage suffisamment complexe, toute structure peut admettre une \u00e9limination des quantificateurs, malheureusement cette extension du langage peut nous \u00e9loigner des ph\u00e9nom\u00e8nes alg\u00e9briques. Nous allons nous int\u00e9resser \u00e0 l\u2019 \u00e9limination des quantificateurs pour des paires de corps. Dans le cas des paires de corps alg\u00e9briquement clos et des paires denses de corps r\u00e9el clos, on obtient une \u00e9limination en ajoutant au langage des pr\u00e9dicats ayant une signification alg\u00e9brique: on peut les exprimer en disant que pour deux ensembles alg\u00e9briques <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001865X_inline1\" \/> et <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001865X_inline2\" \/> donn\u00e9s, il existe des points du sous-corps rationnels pour <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001865X_inline1\" \/> et pas pour <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S002248120001865X_inline2\" \/>, ou qu\u2019 un ensemble semi-alg\u00e9brique donn\u00e9 a des points rationnels sur le sous-corps. Robinson avait d\u00e9j\u00e0 abord\u00e9 de fa\u00e7on informelle le cas des paires denses de corps r\u00e9el-clos (cf. [Ro 2, p. 198]). Partant du langage des paires de corps ordonn\u00e9s, enrichi de symboles de relations correspondant \u00e0 l\u2019 ind\u00e9pendance alg\u00e9brique, il proposait d\u2019 ajouter pas \u00e0 pas des fonctions de Skolem Herbrand pour faire disparaitre les quantificateurs existentiels des formules, mais sans pr\u00e9ciser le langage obtenu. Ici nous approchons le probl\u00e8me diff\u00e9remment en explicitant d\u00e8s le d\u00e9part le langage utilis\u00e9.<\/jats:p><jats:p>Gr\u00e2ce \u00e0 ces r\u00e9sultats nous pourrons \u00e9tudier le cas des paires s\u00e9par\u00e9es de corps r\u00e9els clos ainsi que des paires de corps valu\u00e9s henseliens. En \u00e9largissant la d\u00e9finition d\u2019 ensemble alg\u00e9brique \u00e0 tous les symboles du langage, les pr\u00e9dicats relationnels ajout\u00e9s ont la m\u00eame signification que dans le cas des paires de corps alg\u00e9briquement clos.<\/jats:p><jats:p>En comparant les techniques employ\u00e9es ici avec celles d\u00e9j\u00e0 utilis\u00e9es dans [K], [B], [D 1] et [L], on remarque qu\u2019il est possible de tra\u00eeter une grande partie de l\u2019 \u00e9tude (compl\u00e9tude, mod\u00e8le compl\u00e9tude, \u00e9limination des quantificateurs) des paires de corps alg\u00e9briquement clos, r\u00e9el-clos ou henseliens en se basant sur des prolongements d\u2019 isomorphismes entre sous-structures d\u00e9nombrables o\u00f9 l\u2019 une des deux est contenue dans une structure <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>-satur\u00e9e.<\/jats:p>","DOI":"10.2307\/2275850","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:55:05Z","timestamp":1146956105000},"page":"548-562","source":"Crossref","is-referenced-by-count":1,"title":["\u00c9limination des quantificateurs dans des paires de corps"],"prefix":"10.1017","volume":"60","author":[{"given":"G.","family":"Leloup","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120001865X_ref018","doi-asserted-by":"publisher","DOI":"10.4064\/fm-47-2-179-204"},{"key":"S002248120001865X_ref015","volume-title":"Lectures on formally real fields","volume":"22","author":"Prestel","year":"1975"},{"key":"S002248120001865X_ref013","unstructured":"[M] Macintyre A. 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