Algebra & Number Theory

On the minimal ramification problem for semiabelian groups

Hershy Kisilevsky, Danny Neftin, and Jack Sonn

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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 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

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

Received: 20 December 2009
Revised: 24 June 2010
Accepted: 1 August 2010
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Galois group nilpotent group ramified primes wreath product semiabelian group


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.

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