{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,31]],"date-time":"2025-12-31T16:43:49Z","timestamp":1767199429298,"version":"build-2238731810"},"reference-count":10,"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":9415,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1988,6]]},"abstract":"\n Let\n K<\/jats:italic>\n be an algebraic number field and\n \n I\n K<\/jats:sub>\n <\/jats:italic>\n the ring of algebraic integers in\n K<\/jats:italic>\n . *\n K<\/jats:italic>\n and *\n \n I\n K<\/jats:sub>\n <\/jats:italic>\n denote enlargements of\n K<\/jats:italic>\n and\n \n I\n K<\/jats:sub>\n <\/jats:italic>\n respectively. Let\n x<\/jats:italic>\n \u0404 *\n K<\/jats:italic>\n \u2013\n K<\/jats:italic>\n . In this paper, we are concerned with algebraic extensions of\n K(x)<\/jats:italic>\n within *\n K<\/jats:italic>\n . For each\n x<\/jats:italic>\n \u0404 *\n K<\/jats:italic>\n \u2013\n K<\/jats:italic>\n and each natural number\n \n d, Y\n K<\/jats:sub>\n (x,d)\n <\/jats:italic>\n is defined to be the number of algebraic extensions of\n K(x)<\/jats:italic>\n of degree\n d<\/jats:italic>\n within *\n K<\/jats:italic>\n .\n x<\/jats:italic>\n \u0404 *\n K<\/jats:italic>\n \u2013\n K<\/jats:italic>\n is called a Hilbertian element if\n \n Y\n K<\/jats:sub>\n (x,d)\n <\/jats:italic>\n = 0 for all\n d<\/jats:italic>\n \u0404 N,\n d<\/jats:italic>\n > 1; in other words,\n K(x)<\/jats:italic>\n has no algebraic extension within *\n K<\/jats:italic>\n . In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural number\n \u03c9<\/jats:italic>\n , 2\n \n \u03c9<\/jats:italic>\n <\/jats:sup>\n P<\/jats:italic>\n \n \u03c9<\/jats:italic>\n <\/jats:sub>\n and 2\n \n \u03c9<\/jats:italic>\n <\/jats:sup>\n (\n \u03c9<\/jats:italic>\n 3<\/jats:sup>\n + 1) are Hilbertian elements in\n *Q<\/jats:bold>\n , where p\n \n \u03c9<\/jats:italic>\n <\/jats:sub>\n is the\n \u03c9<\/jats:italic>\n th prime number.\n <\/jats:p>","DOI":"10.2307\/2274519","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:25:43Z","timestamp":1146939943000},"page":"470-480","source":"Crossref","is-referenced-by-count":3,"title":["Algebraic extensions in nonstandard models and Hilbert's irreducibility theorem"],"prefix":"10.1017","volume":"53","author":[{"given":"Masahiro","family":"Yasumoto","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200028413_bib006","first-page":"180","article-title":"Diophantine equations with unknown prime numbers","volume":"158","author":"Sprind\u017euk","year":"1981","journal-title":"Trudy Ordena Lenina Matematiceskogo Instituta imeni V. 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