Skip to main content Accessibility help
×
Home

ON THE DISTRIBUTION OF GALOIS GROUPS

  • Rainer Dietmann (a1)

Abstract

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.

Copyright

References

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

MSC classification

ON THE DISTRIBUTION OF GALOIS GROUPS

  • Rainer Dietmann (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed