{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T14:46:03Z","timestamp":1779893163088,"version":"3.53.1"},"reference-count":32,"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:p>The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the theory of finite fields are exactly the perfect PAC fields with absolute Galois group isomorphic to <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009361_inline1\"\/>, and gave elementary invariants for their first order theory, thereby proving the decidability of the theory of finite fields. Ax's results were then extended to a larger class of PAC fields by M. Jarden and U. Kiehne [21], and Jarden [19]. The final word on theories of PAC fields was given by G. Cherlin, L. van den Dries and A. Macintyre [10], see also results by Ju. Ershov [13], [14]. Let <jats:italic>K<\/jats:italic> be a PAC field. Then the elementary theory of <jats:italic>K<\/jats:italic> is entirely determined by the following data:<\/jats:p><jats:p>\u2022 The isomorphism type of the field of absolute numbers of <jats:italic>K<\/jats:italic> (the subfield of <jats:italic>K<\/jats:italic> of elements algebraic over the prime field).<\/jats:p><jats:p>\u2022 The degree of imperfection of <jats:italic>K<\/jats:italic>.<\/jats:p><jats:p>\u2022 The first-order theory, in a suitable <jats:italic>\u03c9<\/jats:italic>-sorted language, of the inverse system of Galois groups <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009361_inline2\"\/><jats:italic>al(L\/K)<\/jats:italic> where <jats:italic>L<\/jats:italic> runs over all finite Galois extensions of <jats:italic>K<\/jats:italic>.<\/jats:p><jats:p>They also showed that the theory of PAC fields is undecidable, by showing that any graph can be encoded in the absolute Galois group of some PAC field. It turns out that the absolute Galois group controls much of the behaviour of the PAC fields. I will give below some examples illustrating this phenomenon.<\/jats:p>","DOI":"10.2178\/jsl\/1190150143","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:14:26Z","timestamp":1197573266000},"page":"957-996","source":"Crossref","is-referenced-by-count":21,"title":["Properties of forking in \u03c9-free pseudo-algebraically closed fields"],"prefix":"10.1017","volume":"67","author":[{"given":"Zo\u00e9","family":"Chatzidakis","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200009361_ref029","volume-title":"Algebra","author":"Lang","year":"1984"},{"key":"S0022481200009361_ref027","volume-title":"Introduction to algebraic geometry","author":"Lang","year":"1973"},{"key":"S0022481200009361_ref024","first-page":"822","volume":"66","author":"Kim","year":"2001","journal-title":"Simplicity, and stability in there"},{"key":"S0022481200009361_ref031","article-title":"Introduction to profinite groups and Galois cohomology","volume":"24","author":"Ribes","year":"1970","journal-title":"Queen's papers in pure and applied mathematics"},{"key":"S0022481200009361_ref021","doi-asserted-by":"publisher","DOI":"10.1007\/BF01425568"},{"key":"S0022481200009361_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-0072(98)00021-9"},{"key":"S0022481200009361_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-07216-5"},{"key":"S0022481200009361_ref014","doi-asserted-by":"publisher","DOI":"10.1007\/BF01669109"},{"key":"S0022481200009361_ref013","first-page":"510","article-title":"Regularly closed fields","volume":"21","author":"Ershov","year":"1980","journal-title":"Soviet Math. 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