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Minus class groups of the fields
of the $l$-th roots of unity


Author: René Schoof
Journal: Math. Comp. 67 (1998), 1225-1245
MSC (1991): Primary 11R18, 11R29, 11R34
DOI: https://doi.org/10.1090/S0025-5718-98-00939-9
MathSciNet review: 1458225
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Abstract: We show that for any prime number $l>2$ the minus class group of the field of the $l$-th roots of unity $\overline{\mathbf{Q}_p} (\zeta _l)$ admits a finite free resolution of length 1 as a module over the ring $\widehat{\mathbf{Z}} [G]/(1+\iota)$. Here $\iota$ denotes complex conjugation in $G={\textup{Gal}}( \overline{\mathbf{Q}_p} (\zeta _l)/\overline{\mathbf{Q}_p} )\cong(\mathbf{Z} /l\mathbf{Z} )^*$. Moreover, for the primes $l\le 509$ we show that the minus class group is cyclic as a module over this ring. For these primes we also determine the structure of the minus class group.


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Additional Information

René Schoof
Affiliation: Dipartimento di Matematica, $2^{a}$ Università di Roma “Tor Vergata", I-00133 Rome, Italy
Email: schoof@wins.uva.nl

DOI: https://doi.org/10.1090/S0025-5718-98-00939-9
Keywords: Cyclotomic fields, class groups, cohomology of groups
Received by editor(s): March 28, 1994
Received by editor(s) in revised form: December 2, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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