Computation of the $p$-part of the ideal class group of certain real abelian fields
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Abstract:
Under Greenberg’s conjecture, we give an efficient method to compute the $p$-part of the ideal class group of certain real abelian fields by using cyclotomic units, Gauss sums and prime numbers. As numerical examples, we compute the $p$-part of the ideal class group of the maximal real subfield of $\mathbf {Q}(\sqrt {-f},\zeta _{p^{n+1}})$ in the range $1 <f<200$ and $5 \le p <100000$. In order to explain our method, we show an example whose ideal class group is not cyclic.References
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Additional Information
- Hiroki Sumida-Takahashi
- Affiliation: Faculty and School of Engineering, The University of Tokushima, 2-1 Minamijosanjima-cho, Tokushima 770-8506, Japan
- Email: hiroki@pm.tokushima-u.ac.jp
- Received by editor(s): September 7, 2005
- Received by editor(s) in revised form: January 20, 2006
- Published electronically: January 5, 2007
- Additional Notes: This work was partially supported by the Grants-in-Aid for Encouragement of Young Scientists (No. 16740019) from Japan Society for the Promotion of Science.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 76 (2007), 1059-1071
- MSC (2000): Primary 11R23, 11R70
- DOI: https://doi.org/10.1090/S0025-5718-07-01926-6
- MathSciNet review: 2291850