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Mathematics of Computation

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Computing the torsion of the $p$-ramified module of a number field


Authors: Frédéric Pitoun and Firmin Varescon
Journal: Math. Comp. 84 (2015), 371-383
MSC (2010): Primary 11R23, 11R37, 11Y40
DOI: https://doi.org/10.1090/S0025-5718-2014-02838-X
Published electronically: June 24, 2014
MathSciNet review: 3266966
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Abstract: We fix a prime number $p$ and a number field $K$, and denote by $M$ the maximal abelian $p$-extension of $K$ unramified outside $p$. Our aim is to study the $\mathbb {Z}_p$-module $\mathfrak {X}=\mathrm {Gal}(M/K)$ and to give a method to effectively compute its structure as a $\mathbb {Z}_p$-module. We also give numerical results, for real quadratic fields, cubic fields and quintic fields, together with their interpretations via Cohen-Lenstra heuristics.


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Additional Information

Frédéric Pitoun
Affiliation: 27 Avenue du 8 mai 1945, 11400 Castelnaudary, France
Email: frederic.pitoun@free.fr

Firmin Varescon
Affiliation: Laboratoire de mathématiques de Besançon, CNRS UMR 6623, Université de Franche Comté, 16 Route de Gray, 25020 Besançon Cédex, France
Email: firmin.varescon@univ-fcomte.fr

Received by editor(s): April 10, 2012
Received by editor(s) in revised form: February 13, 2013, April 4, 2013, and May 3, 2013
Published electronically: June 24, 2014
Article copyright: © Copyright 2014 American Mathematical Society