{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T05:47:58Z","timestamp":1785304078847,"version":"3.55.0"},"reference-count":24,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2014,5,1]],"date-time":"2014-05-01T00:00:00Z","timestamp":1398902400000},"content-version":"unspecified","delay-in-days":120,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2014]]},"abstract":"Abstract<\/jats:title>Computational Galois theory, in particular the problem of computing the Galois group of a given polynomial, is a very old problem. Currently, the best algorithmic solution is Stauduhar\u2019s method. Computationally, one of the key challenges in the application of Stauduhar\u2019s method is to find, for a given pair of groups $H<G$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, a $G$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-relative $H$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-invariant, that is a multivariate polynomial $F$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> that is $H$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-invariant, but not $G$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>-invariant. While generic, theoretical methods are known to find such $F$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, in general they yield impractical answers. We give a general method for computing invariants of large degree which improves on previous known methods, as well as various special invariants that are derived from the structure of the groups. We then apply our new invariants to the task of computing the Galois groups of polynomials over the rational numbers, resulting in the first practical degree independent algorithm.<\/jats:p>","DOI":"10.1112\/s1461157013000302","type":"journal-article","created":{"date-parts":[[2014,5,19]],"date-time":"2014-05-19T09:10:35Z","timestamp":1400490635000},"page":"141-158","source":"Crossref","is-referenced-by-count":19,"title":["Computation of Galois groups of rational polynomials"],"prefix":"10.1112","volume":"17","author":[{"given":"Claus","family":"Fieker","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J\u00fcrgen","family":"Kl\u00fcners","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2014,5,1]]},"reference":[{"key":"S1461157013000302_r6","volume-title":"Encyclopaedia of mathematical sciences. Invariant theory and algebraic transformation groups","author":"Derksen","year":"2002"},{"key":"S1461157013000302_r11","unstructured":"11. K. Gei\u00dfler , \u2018Berechnung von Galoisgruppen \u00fcber Zahl- und Funktionenk\u00f6rpern\u2019, PhD Thesis, Technische Universit\u00e4t Berlin, 2003."},{"key":"S1461157013000302_r18","doi-asserted-by":"publisher","DOI":"10.1006\/jsco.2000.0378"},{"key":"S1461157013000302_r15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2012.05.010"},{"key":"S1461157013000302_r22","unstructured":"22. L. Soicher , \u2018The computation of Galois groups\u2019, M. Comp. Sci. Thesis, Concordia University, Montreal, 1981. http:\/\/www.maths.qmul.ac.uk\/\u223cleonard\/mcompsc_soicher.pdf."},{"key":"S1461157013000302_r10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2011.11.006"},{"key":"S1461157013000302_r12","doi-asserted-by":"publisher","DOI":"10.1006\/jsco.2000.0377"},{"key":"S1461157013000302_r16","volume-title":"Computational Methods for Representations of Groups and Algebras, Euroconference in Essen, April 1\u20135 1997","author":"Kemper","year":"1997"},{"key":"S1461157013000302_r1","doi-asserted-by":"publisher","DOI":"10.1051\/ita:1999106"},{"key":"S1461157013000302_r9","unstructured":"9. Y. Eichenlaub , \u2018Probl\u00e8mes effectifs de th\u00e9orie de Galois en degr\u00e9s 8 \u00e0 11\u2019, Th\u00e8se, Universit\u00e9 Bordeaux 1, 1996."},{"key":"S1461157013000302_r4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02945-9"},{"key":"S1461157013000302_r7","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0731-3"},{"key":"S1461157013000302_r14","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1987-0878706-1"},{"key":"S1461157013000302_r19","doi-asserted-by":"publisher","DOI":"10.1112\/S1461157000000851"},{"key":"S1461157013000302_r5","doi-asserted-by":"publisher","DOI":"10.1112\/S1461157000000115"},{"key":"S1461157013000302_r2","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/pdn039"},{"key":"S1461157013000302_r13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01165834"},{"key":"S1461157013000302_r8","doi-asserted-by":"publisher","DOI":"10.2140\/ant.2013.7.1325"},{"key":"S1461157013000302_r20","first-page":"173","volume-title":"Computer algebra. Symbolic and algebraic computation","author":"Loos","year":"1982"},{"key":"S1461157013000302_r24","first-page":"124","volume-title":"ANTS 2006","author":"Yokoyama","year":"2006"},{"key":"S1461157013000302_r21","first-page":"109","article-title":"Verwendung von Untergruppenleitern zur Bestimmung von Doppelnebenklassen","volume":"31","author":"Schmalz","year":"1990","journal-title":"Bayreuther Math. Schriften"},{"key":"S1461157013000302_r23","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1973-0327712-4"},{"key":"S1461157013000302_r3","first-page":"749","article-title":"Symmetric functions, m-sets, and Galois groups","volume":"63","author":"Casperson","year":"1994","journal-title":"Math. Comput."},{"key":"S1461157013000302_r17","unstructured":"17. J. Kl\u00fcners , \u2018\u00dcber die Berechnung von Automorphismen und Teilk\u00f6rpern algebraischer Zahlk\u00f6rper\u2019, PhD\u00a0TU-Berlin (1997)."}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157013000302","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,22]],"date-time":"2019-04-22T22:04:38Z","timestamp":1555970678000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157013000302\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":24,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2014]]}},"alternative-id":["S1461157013000302"],"URL":"https:\/\/doi.org\/10.1112\/s1461157013000302","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014]]}}}