Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-15T21:41:31.918Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  21 October 2011

Rainer Dietmann*
Department of Mathematics, Royal Holloway, University of London, Egham, TW20 0EX, U.K. (email:
Get access


Let G be a subgroup of the symmetric group Sn, and let δG=∣Sn/G−1 where ∣Sn/G∣ is the index of G in Sn. Then there are at most On(Hn−1+δG) monic integer polynomials of degree n that have Galois group G and height not exceeding H, so there are only a “few” polynomials having a “small” Galois group.

Research Article
Copyright © University College London 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


[1]Bombieri, E. and Pila, J., The number of integral points on arcs and ovals. Duke Math. J. 59 (1989), 337357.CrossRefGoogle Scholar
[2]Browning, T. D. and Heath-Brown, D. R., Plane curves in boxes and equal sums of two powers. Math. Z. 251 (2005), 233247.CrossRefGoogle Scholar
[3]Chavdarov, N., The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy. Duke Math. J. 87 (1997), 151180.CrossRefGoogle Scholar
[4]Chela, R., Reducible polynomials. J. Lond. Math. Soc. 38 (1963), 183188.CrossRefGoogle Scholar
[5]Cohen, S. D., The distribution of the Galois groups of integral polynomials. Illinois J. Math. 23 (1979), 135152.CrossRefGoogle Scholar
[6]Davis, S., Duke, W. and Sun, X., Probabilistic Galois theory of reciprocal polynomials. Expo. Math. 16 (1998), 263270.Google Scholar
[7]Dietmann, R., Probabilistic Galois theory for quartic polynomials. Glasg. Math. J. 48(3) (2006), 553556.CrossRefGoogle Scholar
[8]Dixon, J. D. and Mortimer, B., Permutation Groups (Graduate Texts in Mathematics 163), Springer (New York, 1996).CrossRefGoogle Scholar
[9]Gallagher, P. X., The Large Sieve and Probabilistic Galois Theory (Proceedings of Symposia in Pure Mathematics 23), American Mathematical Society (Providence, RI, 1973), 91101.Google Scholar
[10]Heath-Brown, D. R., The density of rational points on curves and surfaces. Ann. of Math. (2) 155 (2002), 553598.CrossRefGoogle Scholar
[11]Heath-Brown, D. R., Counting rational points on algebraic varieties. In Analytic Number Theory (Lecture Notes in Mathematics 1891), Springer (New York, 2006), 5195.CrossRefGoogle Scholar
[12]Hering, H., Seltenheit der Gleichungen mit Affekt bei linearem Parameter. Math. Ann. 186 (1970), 263270.CrossRefGoogle Scholar
[13]Hering, H., Über Koeffizientenbeschränkungen affektloser Gleichungen. Math. Ann. 195 (1972), 121136.CrossRefGoogle Scholar
[14]Klüners, J. and Malle, G., Explicit Galois realization of transitive groups of degree up to 15. J. Symbolic Comput. 30 (2000), 675716.CrossRefGoogle Scholar
[15]Kowalski, E., The large sieve, monodromy and zeta functions of curves. J. Reine Angew. Math. 601 (2006), 2969.Google Scholar
[16]Lefton, P., Galois resolvents of permutation groups. Amer. Math. Monthly 84 (1977), 642644.CrossRefGoogle Scholar
[17]Lefton, P., On the Galois groups of cubics and trinomials. Acta Arith. 35 (1979), 239246.CrossRefGoogle Scholar
[18]Marden, M., Geometry of polynomials, 2nd edn. (Mathematical Surveys 3), American Mathematical Society (Providence, RI, 1966).Google Scholar
[19]Rivin, I., Walks on groups, counting reducible matrices, polynomials, and surface and free group automorphisms. Duke Math. J. 142 (2008), 353379.CrossRefGoogle Scholar
[20]van der Waerden, B. L., Die Seltenheit der reduziblen Gleichungen und die Gleichungen mit Affekt. Monatsh. Math. 43 (1936), 137147.Google Scholar
[21]Zarhin, Y. G., Very simple 2-adic representations and hyperelliptic Jacobians. Mosc. Math. J. 2 (2002), 403451.CrossRefGoogle Scholar
[22]Zywina, D., Hilbert’s irreducibility theorem and the larger sieve. arXiv:1011.6465v1 [math.NT].Google Scholar