Rocky Mountain Journal of Mathematics

The discriminant of abelian number fields

Victor Bautista-Ancona and Jose Uc-Kuk

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


For an abelian number field $K$, the discriminant can be obtained from the conductor~$m$ of~$K$, the degree of~$K$ over $\mathbb {Q}$, and the degrees of extensions $K\cdot \mathbb {Q}(\zeta _{m/p^{\alpha }})/\mathbb {Q}(\zeta _{m/p^{\alpha }})$, where $p$ runs through the set of primes that divide $m$, and $p^{\alpha }$ is the greatest power that divides~$m$. In this paper, we give a formula for computing the discriminant of any abelian number field.

Article information

Rocky Mountain J. Math., Volume 47, Number 1 (2017), 39-52.

First available in Project Euclid: 3 March 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R18: Cyclotomic extensions 11R29: Class numbers, class groups, discriminants

Cyclotomic fields abelian number fields conductors discriminant


Bautista-Ancona, Victor; Uc-Kuk, Jose. The discriminant of abelian number fields. Rocky Mountain J. Math. 47 (2017), no. 1, 39--52. doi:10.1216/RMJ-2017-47-1-39.

Export citation


  • José Othon Dantas Lopes, The discriminant of subfields of $\mathbb{Q}(\zeta_{2^r})$, J. Alg. Appl. 2 (2003), 463–469.
  • J. Carmelo Interlando, José Othon Dantas Lopes and Trajano Pires da N. Neto, The discriminant of abelian numbers fields, J. Alg. Appl. 5 (2006), 35–41.
  • Trajano Pires Da N. Neto, J. Carmelo Interlando and José Othon Dantas Lopes, On computing discriminants of subfields of $\mathbb{Q}(\zeta_{p^r})$, J. Num. Th. 96 (2002), 319–325.
  • L.C. Washington, Introduction to cyclotomic fields, Second edition, Grad. Texts Math. 83, Springer-Verlag, New York, 1997.