Skip to main content


  • Lior Bary-Soroker (a1) and Tomer M. Schlank (a2)

The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group $G$ , let $m(G)$ be the minimal integer $m$ for which there exists a $G$ -Galois extension $N/\mathbb{Q}$ that is ramified at exactly $m$ primes (including the infinite one). So, the problem is to compute or to bound $m(G)$ .

In this paper, we bound the ramification of extensions $N/\mathbb{Q}$ obtained as a specialization of a branched covering $\unicode[STIX]{x1D719}:C\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$ . This leads to novel upper bounds on $m(G)$ , for finite groups $G$ that are realizable as the Galois group of a branched covering. Some instances of our general results are:

$$\begin{eqnarray}1\leqslant m(S_{k})\leqslant 4\quad \text{and}\quad n\leqslant m(S_{k}^{n})\leqslant n+4,\end{eqnarray}$$
for all $n,k>0$ . Here $S_{k}$ denotes the symmetric group on $k$ letters, and $S_{k}^{n}$ is the direct product of $n$ copies of $S_{k}$ . We also get the correct asymptotic of $m(G^{n})$ , as $n\rightarrow \infty$ for a certain class of groups $G$ .

Our methods are based on sieve theory results, in particular on the Green–Tao–Ziegler theorem on prime values of linear forms in two variables, on the theory of specialization in arithmetic geometry, and on finite group theory.

Hide All
1. Beckmann, S., On extensions of number fields obtained by specializing branched coverings, J. Reine Angew. Math. 419 (1991), 2753.
2. Boston, N. and Markin, N., The fewest primes ramified in a G-extension of ℚ, Ann. Sci. Math. Québec 33(2) (2009), 145154.
3. De Witt, M., Minimal ramification and the inverse Galois problem over the rational function field F p (t), J. Number Theory 143 (2014), 6281.
4. Fried, M. D. and Jarden, M., Field Arithmetic, 3rd edn, Revised by Jarden, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Volume 11 (Springer, Berlin, 2008).
5. Green, B., Pop, F. and Roquette, P., On Rumely’s local-global principle, Jahresber. Dtsch. Math.-Ver. 97(2) (1995), 4374.
6. Green, B. and Tao, T., Linear equations in primes, Ann. of Math. (2) 171(3) (2010), 17531850.
7. Green, B. and Tao, T., The Möbius function is strongly orthogonal to nilsequences, Ann. of Math. (2) 175(2) (2012), 541566.
8. Green, B., Tao, T. and Ziegler, T., An inverse theorem for the Gowers U s+1[N]-norm, Ann. of Math. (2) 176(2) (2012), 12311372.
9. Halberstam, H. and Richert, H.-E., Sieve Methods, London Mathematical Society Monographs, Volume 4 (Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1974).
10. Harbater, D., Abhyankar’s conjecture on Galois groups over curves, Invent. Math. 117(1) (1994), 125.
11. Harpaz, Y., Skorobogatov, A. N. and Wittenberg, O., The Hardy-Littlewood conjecture and rational points, Compositio Mathematica 150(12) (2014), 20952111.
12. Hoelscher, J. L., Galois extensions ramified only at one prime, J. Number Theory 129(2) (2009), 418427.
13. Jones, J. W. and Roberts, D. P., Number fields ramified at one prime, in Algorithmic Number Theory, Vol. 5011, Lecture Notes in Computational Science, pp. 226239 (Springer, Berlin, 2008).
14. Khare, C., Larsen, M. and Savin, G., Functoriality and the inverse Galois problem, Compos. Math. 144(3) (2008), 541564.
15. Kisilevsky, H., Neftin, D. and Sonn, J., On the minimal ramification problem for semiabelian groups, Algebra Number Theory 4(8) (2010), 10771090.
16. Kisilevsky, H. and Sonn, J., On the minimal ramification problem for -groups, Compos. Math. 146(3) (2010), 599606.
17. Legrand, F., Specialization results and ramification conditions, Israel Journal of Mathematics 214(2) (2016), 621650.
18. Malle, G. and Heinrich Matzat, B., Inverse Galois Theory, Springer Monographs in Mathematics (Springer, Berlin, 1999).
19. Malle, G. and Roberts, D. P., Number fields with discriminant ± 2 a 3 b and Galois group A n or S n , LMS J. Comput. Math. 8 (2005), 80101 (electronic).
20. Markin, N. and Ullom, S. V., Minimal ramification in nilpotent extensions, Pacific J. Math. 253(1) (2011), 125143.
21. Milne, J. S., Étale Cohomology, Princeton Mathematical Series, Volume 33 (Princeton University Press, Princeton, NJ, 1980).
22. Neukirch, J., Schmidt, A. and Wingberg, K., Cohomology of Number Fields, 2nd edn, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Volume 323 (Springer, Berlin, 2008).
23. Nomura, A., Notes on the minimal number of ramified primes in some l-extensions of Q , Arch. Math. (Basel) 90(6) (2008), 501510.
24. Plans, B., On the minimal number of ramified primes in some solvable extensions of ℚ, Pacific J. Math. 215(2) (2004), 381391.
25. Raynaud, M., Revêtements de la droite affine en caractéristique p > 0 et conjecture d’Abhyankar, Invent. Math. 116(1–3) (1994), 425462.
26. Serre, J.-P., Topics in Galois Theory, 2nd edn, Research Notes in Mathematics, (A K Peters, Ltd., 2008).
27.The Stacks Project Authors. stacks project., 2015.
28. Völklein, H., Groups as Galois Groups, An Introduction, Cambridge Studies in Advanced Mathematics, Volume 53 (Cambridge University Press, Cambridge, 1996).
29. Wiese, G., On projective linear groups over finite fields as Galois groups over the rational numbers, in Modular Forms on Schiermonnikoog, pp. 343350 (Cambridge University Press, Cambridge, 2008).
30. Zywina, D., The inverse Galois problem for PSL2(F p ), Duke Mathematical Journal 164(12) (2015), 22532292.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Institute of Mathematics of Jussieu
  • ISSN: 1474-7480
  • EISSN: 1475-3030
  • URL: /core/journals/journal-of-the-institute-of-mathematics-of-jussieu
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 17 *
Loading metrics...

Abstract views

Total abstract views: 59 *
Loading metrics...

* Views captured on Cambridge Core between 18th June 2018 - 22nd August 2018. This data will be updated every 24 hours.