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Mathematics of Computation

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Computation of the $p$-part of the ideal class group of certain real abelian fields

Author: Hiroki Sumida-Takahashi
Journal: Math. Comp. 76 (2007), 1059-1071
MSC (2000): Primary 11R23, 11R70
Published electronically: January 5, 2007
MathSciNet review: 2291850
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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.

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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

Keywords: Ideal class group, Iwasawa invariant, abelian field, Greenberg’s conjecture
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.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.