{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T16:31:21Z","timestamp":1767198681289,"version":"build-2238731810"},"reference-count":0,"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":13798,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1976,6]]},"abstract":"<jats:p>Since the late 1940's model theory has found numerous applications to algebra. I would like to indicate some of the points of contact between model theoretic methods and strictly algebraic concerns by means of a few concrete examples and typical applications.<\/jats:p>\n                  <jats:p>\n                    <jats:bold>\u00a71. The Lefschetz principle.<\/jats:bold>\n                    Algebraic geometry has proved to be a fruitful source of model theoretic ideas. What exactly is algebraic geometry? We consider a field\n                    <jats:italic>K<\/jats:italic>\n                    , and let\n                    <jats:italic>\n                      K\n                      <jats:sup>n<\/jats:sup>\n                    <\/jats:italic>\n                    be the set of\n                    <jats:italic>n<\/jats:italic>\n                    -tuples (\n                    <jats:italic>a<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    \u2026\n                    <jats:italic>\n                      a\n                      <jats:sub>n<\/jats:sub>\n                    <\/jats:italic>\n                    ) with coordinates\n                    <jats:italic>\n                      a\n                      <jats:sub>i<\/jats:sub>\n                    <\/jats:italic>\n                    in\n                    <jats:italic>K<\/jats:italic>\n                    .\n                    <jats:italic>\n                      K\n                      <jats:sup>n<\/jats:sup>\n                    <\/jats:italic>\n                    is called\n                    <jats:italic>affine n-space<\/jats:italic>\n                    over\n                    <jats:italic>K<\/jats:italic>\n                    . Fix polynomials\n                    <jats:italic>p<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    \u2026,\n                    <jats:italic>\n                      p\n                      <jats:sub>k<\/jats:sub>\n                    <\/jats:italic>\n                    in\n                    <jats:italic>K<\/jats:italic>\n                    [\n                    <jats:italic>x<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    , \u2026,\n                    <jats:italic>\n                      x\n                      <jats:sub>n<\/jats:sub>\n                    <\/jats:italic>\n                    ] and define\n                  <\/jats:p>\n                  <jats:p>\n                    <jats:disp-formula>\n                      <jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200051616_Uequ1.png\"\/>\n                    <\/jats:disp-formula>\n                  <\/jats:p>\n                  <jats:p>\n                    that is\n                    <jats:italic>V<\/jats:italic>\n                    (\n                    <jats:italic>p<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    \u2026,\n                    <jats:italic>\n                      p\n                      <jats:sub>k<\/jats:sub>\n                    <\/jats:italic>\n                    ) is the locus of common zeroes of the\n                    <jats:italic>\n                      p\n                      <jats:sub>i<\/jats:sub>\n                    <\/jats:italic>\n                    <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"png\" xlink:href=\"S0022481200051616_inline1.png\"\/>\n                    in\n                    <jats:italic>\n                      K\n                      <jats:sub>n<\/jats:sub>\n                    <\/jats:italic>\n                    . We call\n                    <jats:italic>V<\/jats:italic>\n                    (\n                    <jats:italic>\n                      P\n                      <jats:sub>i<\/jats:sub>\n                    <\/jats:italic>\n                    \u2026,\n                    <jats:italic>\n                      P\n                      <jats:sub>k<\/jats:sub>\n                    <\/jats:italic>\n                    ) the\n                    <jats:italic>algebraic variety<\/jats:italic>\n                    determined by\n                    <jats:italic>p<\/jats:italic>\n                    <jats:sub>1<\/jats:sub>\n                    , \u2026,\n                    <jats:italic>\n                      p\n                      <jats:sub>k<\/jats:sub>\n                    <\/jats:italic>\n                    . With this terminology we may say:\n                  <\/jats:p>\n                  <jats:p>\n                    Algebraic geometry is the study of algebraic varieties defined over an arbitrary field\n                    <jats:italic>K<\/jats:italic>\n                    . This definition lacks both rigor and accuracy, and we will indicate below how it may be improved.\n                  <\/jats:p>\n                  <jats:p>\n                    So far we have placed no restrictions on the base field\n                    <jats:italic>K<\/jats:italic>\n                    . Following Weil [4] it is convenient to start with a so-called \u201cuniversal domain\u201d; in other words take\n                    <jats:italic>K<\/jats:italic>\n                    to be algebraically closed and of infinite transcendence degree over the prime field. Any particular field can of course be embedded in such a universal domain.\n                  <\/jats:p>","DOI":"10.2307\/2272254","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:41:13Z","timestamp":1146937273000},"page":"537-545","source":"Crossref","is-referenced-by-count":7,"title":["Model Theoretic Algebra"],"prefix":"10.1017","volume":"41","author":[{"given":"G. L.","family":"Cherlin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"container-title":["The Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200051616","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,23]],"date-time":"2023-03-23T23:10:13Z","timestamp":1679613013000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200051616\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1976,6]]},"references-count":0,"aliases":["10.1017\/s0022481200051616"],"journal-issue":{"issue":"2","published-print":{"date-parts":[[1976,6]]}},"alternative-id":["S0022481200051616"],"URL":"https:\/\/doi.org\/10.2307\/2272254","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1976,6]]}}}