{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T01:43:51Z","timestamp":1768700631551,"version":"3.49.0"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"236","license":[{"start":{"date-parts":[[2002,3,8]],"date-time":"2002-03-08T00:00:00Z","timestamp":1015545600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>Let <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold k\">\n  <mml:semantics>\n    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n      <mml:mi mathvariant=\"bold\">k<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbf {k}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> be a <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n  <mml:semantics>\n    <mml:mi>p<\/mml:mi>\n    <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula>-adic field. It is well-known that <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold k\">\n  <mml:semantics>\n    <mml:mrow class=\"MJX-TeXAtom-ORD\">\n      <mml:mi mathvariant=\"bold\">k<\/mml:mi>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbf {k}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> has only finitely many extensions of a given finite degree. Krasner has given formulae for the number of extensions of a given degree and discriminant. Following his work, we present an algorithm for the computation of generating polynomials for all extensions <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold upper K slash bold k\">\n  <mml:semantics>\n    <mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"bold\">K<\/mml:mi>\n      <\/mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mo>\/<\/mml:mo>\n      <\/mml:mrow>\n      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n        <mml:mi mathvariant=\"bold\">k<\/mml:mi>\n      <\/mml:mrow>\n    <\/mml:mrow>\n    <mml:annotation encoding=\"application\/x-tex\">\\mathbf {K}\/\\mathbf {k}<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> of a given degree and discriminant.<\/p>","DOI":"10.1090\/s0025-5718-01-01306-0","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:13:53Z","timestamp":1027721633000},"page":"1641-1659","source":"Crossref","is-referenced-by-count":28,"title":["On the computation of all extensions of a \ud835\udc5d-adic field of a given degree"],"prefix":"10.1090","volume":"70","author":[{"given":"Sebastian","family":"Pauli","sequence":"first","affiliation":[]},{"given":"Xavier-Fran\u00e7ois","family":"Roblot","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,3,8]]},"reference":[{"key":"1","unstructured":"[Ba99] C. Batut, K. Belabas, D. Bernardi, H. Cohen, M. Olivier. The Computer Algebra System PARI-GP, Universit\u00e9 Bordeaux I, 1999, \\url{ftp:\/\/megrez.math.u-bordeaux.fr\/pub\/pari\/}"},{"key":"2","series-title":"London Mathematical Society Student Texts","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139171885","volume-title":"Local fields","volume":"3","author":"Cassels, J. W. S.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0521304849"},{"issue":"3-4","key":"3","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1006\/jsco.1996.0126","article-title":"KANT V4","volume":"24","author":"Daberkow, M.","year":"1997","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","first-page":"590","DOI":"10.1215\/S0012-7094-39-00549-1","article-title":"Some properties of polynomial sets of type zero","volume":"5","author":"Sheffer, I. M.","year":"1939","journal-title":"Duke Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0012-7094","issn-type":"print"},{"issue":"1","key":"5","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1006\/jnth.1996.0092","article-title":"De nouveaux invariants num\u00e9riques pour les extensions totalement ramifi\u00e9es de corps locaux","volume":"59","author":"Heiermann, Volker","year":"1996","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"6","first-page":"143","article-title":"Nombre des extensions d\u2019un degr\u00e9 donn\u00e9 d\u2019un corps \ud835\udd2d-adique","author":"Krasner, Marc","year":"1966"},{"issue":"18","key":"7","first-page":"A863--A865","article-title":"Remarques au sujet d\u2019une note de J.-P. Serre: \u201cUne \u2018formule de masse\u2019 pour les extensions totalement ramifi\u00e9es de degr\u00e9 donn\u00e9 d\u2019un corps local\u201d [C. R. Acad. Sci. Paris S\u00e9r. A-B 286 (1978), no. 22, A1031\u2013A1036; MR 80a:12018]: une d\u00e9monstration de la formule de M. Serre \u00e0 partir de mon th\u00e9or\u00e8me sur le nombre des extensions s\u00e9parables d\u2019un corps valu\u00e9 localement compact, qui sont d\u2019un degr\u00e9 et d\u2019une diff\u00e9rente donn\u00e9s","volume":"288","author":"Krasner, Marc","year":"1979","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. A-B","ISSN":"https:\/\/id.crossref.org\/issn\/0151-0509","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","unstructured":"[Or26] \u00d6. Ore, Bemerkungen zur Theorie der Differente, Math. Zeitschr. 25 (1926), pp. 1\u20138.","DOI":"10.1007\/BF01283823"},{"key":"9","unstructured":"[Pa95] P. Panayi, Computation of Leopoldt\u2019s p-adic regulator, PhD thesis, University of East Anglia, 1995, \\url{http:\/\/www.mth.uea.ac.uk\/ h090\/}."},{"key":"10","series-title":"Publications de l'Institut de Math\\'{e}matique de l'Universit\\'{e} de Nancago, VIII","volume-title":"Corps locaux","author":"Serre, Jean-Pierre","year":"1962"},{"issue":"22","key":"11","first-page":"A1031--A1036","article-title":"Une \u201cformule de masse\u201d pour les extensions totalement ramifi\u00e9es de degr\u00e9 donn\u00e9 d\u2019un corps local","volume":"286","author":"Serre, Jean-Pierre","year":"1978","journal-title":"C. R. Acad. Sci. Paris S\\'{e}r. A-B","ISSN":"https:\/\/id.crossref.org\/issn\/0151-0509","issn-type":"print"},{"key":"12","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1017\/S0370164600012281","article-title":"On the reciprocation of certain matrices","volume":"59","author":"Collar, A. R.","year":"1939","journal-title":"Proc. Roy. Soc. 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