## Algebra & Number Theory

### On the minimal ramification problem for semiabelian groups

#### Abstract

It is now known that for any prime $p$ and any finite semiabelian $p$-group $G$, there exists a (tame) realization of $G$ as a Galois group over the rationals $ℚ$ with exactly $d= d(G)$ ramified primes, where $d(G)$ is the minimal number of generators of $G$, which solves the minimal ramification problem for finite semiabelian $p$-groups. We generalize this result to obtain a theorem on finite semiabelian groups and derive the solution to the minimal ramification problem for a certain family of semiabelian groups that includes all finite nilpotent semiabelian groups $G$. Finally, we give some indication of the depth of the minimal ramification problem for semiabelian groups not covered by our theorem.

#### Article information

Source
Algebra Number Theory, Volume 4, Number 8 (2010), 1077-1090.

Dates
Revised: 24 June 2010
Accepted: 1 August 2010
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513729610

Digital Object Identifier
doi:10.2140/ant.2010.4.1077

Mathematical Reviews number (MathSciNet)
MR2832635

Zentralblatt MATH identifier
1221.11218

Subjects
Primary: 11R32: Galois theory
Secondary: 20D15: Nilpotent groups, $p$-groups

#### Citation

Kisilevsky, Hershy; Neftin, Danny; Sonn, Jack. On the minimal ramification problem for semiabelian groups. Algebra Number Theory 4 (2010), no. 8, 1077--1090. doi:10.2140/ant.2010.4.1077. https://projecteuclid.org/euclid.ant/1513729610

#### References

• N. C. Ankeny and S. Chowla, “On the divisibility of the class number of quadratic fields”, Pacific J. Math. 5 (1955), 321–324.
• M. Bhargava, “The density of discriminants of quartic rings and fields”, Ann. of Math. $(2)$ 162:2 (2005), 1031–1063.
• R. Dentzer, “On geometric embedding problems and semiabelian groups”, Manuscripta Math. 86:2 (1995), 199–216.
• D. Harbater, “Galois groups with prescribed ramification”, pp. 35–60 in Arithmetic geometry (Tempe, AZ, 1993), edited by N. Childress and J. W. Jones, Contemp. Math. 174, Amer. Math. Soc., Providence, RI, 1994.
• K. Iwasawa, “A note on class numbers of algebraic number fields”, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257–258.
• M. J. Jacobson, Jr., R. F. Lukes, and H. C. Williams, “An investigation of bounds for the regulator of quadratic fields”, Experiment. Math. 4 (1995), 211–225.
• J. W. Jones and D. P. Roberts, “Number fields ramified at one prime”, pp. 226–239 in Algorithmic number theory, edited by A. J. van der Poorten and A. Stein, Lecture Notes in Comput. Sci. 5011, Springer, Berlin, 2008.
• G. Kaplan and A. Lev, “On the dimension and basis concepts in finite groups”, Comm. Algebra 31:6 (2003), 2707–2717.
• H. Kisilevsky and J. Sonn, “Abelian extensions of global fields with constant local degree”, Math. Res. Lett. 13:4 (2006), 599–605.
• H. Kisilevsky and J. Sonn, “On the minimal ramification problem for $\ell$-groups”, Compos. Math. 146:3 (2010), 599–606.
• J. D. P. Meldrum, Wreath products of groups and semigroups, Pitman Monographs and Surveys in Pure and Applied Mathematics 74, Longman, Harlow, 1995.
• D. Neftin, “On semiabelian $p$-groups”, preprint, 2009.
• B. Plans, “On the minimal number of ramified primes in some solvable extensions of $\mathbb Q$”, Pacific J. Math. 215:2 (2004), 381–391.
• D. Rabayev, Polynomials with roots mod $n$ for all $n$, Master's Thesis, Technion, 2009.
• H. te Riele and H. Williams, “New computations concerning the Cohen–Lenstra heuristics”, Experiment. Math. 12 (2003), 99–113. http://www.emis.de/cgi-bin/MATH-item?1050.11096Zbl 1050.11096