{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T17:08:52Z","timestamp":1760202532741},"reference-count":19,"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":8593,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We say <jats:italic>\u03c6<\/jats:italic> is an \u2200\u2203 sentence if and only if <jats:italic>\u03c6<\/jats:italic> is logically equivalent to a sentence of the form \u2200<jats:italic>x<\/jats:italic>\u2203<jats:italic>y\u03c8<\/jats:italic>(<jats:italic>x, y<\/jats:italic>), where <jats:italic>\u03c8<\/jats:italic>(<jats:italic>x, y<\/jats:italic>) is a quantifier-free formula containing no variables except <jats:italic>x<\/jats:italic> and <jats:italic>y<\/jats:italic>. In this paper we show that there are algorithms to decide whether or not a given \u2200\u2203 sentence is true in (1) an algebraic number field <jats:italic>K<\/jats:italic>, (2) a purely transcendental extension of an algebraic number field <jats:italic>K<\/jats:italic>, (3) every field with characteristic 0, (4) every algebraic number field, (5) every cyclic (abelian, radical) extension field over <jats:bold>Q<\/jats:bold>, and (6) every field.<\/jats:p>","DOI":"10.2307\/2274469","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:36:50Z","timestamp":1146940610000},"page":"1007-1018","source":"Crossref","is-referenced-by-count":4,"title":["Decidable fragments of field theories"],"prefix":"10.1017","volume":"55","author":[{"given":"Shih-Ping","family":"Tung","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025512_ref007","doi-asserted-by":"publisher","DOI":"10.1137\/0214015"},{"key":"S0022481200025512_ref014","doi-asserted-by":"publisher","DOI":"10.3998\/mpub.9690541"},{"key":"S0022481200025512_ref005","volume-title":"A mathematical introduction to logic","author":"Enderton","year":"1972"},{"key":"S0022481200025512_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF01095643"},{"key":"S0022481200025512_ref010","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1959-0112842-7"},{"key":"S0022481200025512_ref019","volume-title":"Modern algebra","volume":"1","author":"van der Waerden","year":"1949"},{"key":"S0022481200025512_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-6101-8"},{"key":"S0022481200025512_ref009","first-page":"98","volume":"14","author":"Robinson","year":"1949","journal-title":"Definability and decision problems in arithmetic"},{"key":"S0022481200025512_ref013","first-page":"211","volume-title":"Journ\u00e9es arithm\u00e9tiques 1980 (lectures, Exeter, 1980","volume":"56","author":"Schinzel","year":"1982"},{"key":"S0022481200025512_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/028.2\/0432534"},{"key":"S0022481200025512_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090173"},{"key":"S0022481200025512_ref008","volume-title":"Algebra","author":"Lang","year":"1965"},{"key":"S0022481200025512_ref011","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1951-0041080-4"},{"key":"S0022481200025512_ref012","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19640101803"},{"key":"S0022481200025512_ref015","first-page":"79","article-title":"A non-standard model for a free-variable fragment of number theory","volume":"12","author":"Shepherdson","year":"1964","journal-title":"Bulletin de l'Acad\u00e9mie Polonaise des Sciences, S\u00e9rie des Sciences Math\u00e9matiques, Astronomiques et Physiques"},{"key":"S0022481200025512_ref016","first-page":"3","article-title":"Achievements and problems in Diophantine approximation theory","volume":"35","author":"Sprindzhuk","year":"1980","journal-title":"Uspekhi Matematicheskikh Nauk"},{"key":"S0022481200025512_ref017","doi-asserted-by":"crossref","first-page":"203","DOI":"10.4099\/math1924.11.203","article-title":"On weak number theories","volume":"11","author":"Tung","year":"1985","journal-title":"Japanese Journal of Mathematics"},{"key":"S0022481200025512_ref018","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/18.3.241"},{"key":"S0022481200025512_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/1970573"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200025512","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T16:25:12Z","timestamp":1558196712000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200025512\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,9]]},"references-count":19,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1990,9]]}},"alternative-id":["S0022481200025512"],"URL":"https:\/\/doi.org\/10.2307\/2274469","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,9]]}}}