Minus class groups of the fields of the $l$-th roots of unity
HTML articles powered by AMS MathViewer
- by René Schoof PDF
- Math. Comp. 67 (1998), 1225-1245 Request permission
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 $G={{Gal}}( \overline {\mathbf {Q}_p} (\zeta _l)/\overline {\mathbf {Q}_p} )\cong (\mathbf {Z} /l\mathbf {Z} )^*$. 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.References
- Bourbaki, N.: Éléments de Mathématique, Algèbre, Hermann, Paris 1970.
- J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
- Cornacchia, P.: Anderson’s module and ideal class groups of abelian fields, J. Number Theory, to appear.
- Cornelius Greither, Class groups of abelian fields, and the main conjecture, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 3, 449–499 (English, with English and French summaries). MR 1182638, DOI 10.5802/aif.1299
- P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
- Kenkichi Iwasawa, A note on ideal class groups, Nagoya Math. J. 27 (1966), 239–247. MR 197438, DOI 10.1017/S0027763000012046
- V. A. Kolyvagin, Euler systems, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 435–483. MR 1106906
- Ernst Eduard Kummer, Collected papers, Springer-Verlag, Berlin-New York, 1975. Volume I: Contributions to number theory; Edited and with an introduction by André Weil. MR 0465760
- Kummer, E.E.: Bestimmung der Anzahl nicht äquivalenter Classen für die aus $\lambda$ten Wurzeln der Einheit gebildeten complexen Zahlen und die idealen Factoren derselben, J. für die reine und angewandte Math. 40, (1850), 93–116. (Coll.Papers 299–322)
- Kummer, E.E.: Mémoire sur la théorie des nombres complexes composés de racines de l’unité et de nombres entiers, J. de math. pures et appl. 16, (1851), 377–498. (Coll.Papers 363-484)
- Kummer, E.E.: Über die Irregularität von Determinanten, Monatsberichte der Kön. Preuß. Ak. der Wiss. zu Berlin, (1853), 194–200. (Coll.Papers 539–545)
- Kummer, E.E.: Über die aus 31sten Wurzeln der Einheit gebildeten complexen Zahlen, Monatsberichte der Kön. Preuß. Ak. der Wiss. zu Berlin, (1870), 755–766. (Coll.Papers 907–918)
- Serge Lang, Cyclotomic fields, Graduate Texts in Mathematics, Vol. 59, Springer-Verlag, New York-Heidelberg, 1978. MR 0485768, DOI 10.1007/978-1-4612-9945-5
- D. H. Lehmer, Prime factors of cyclotomic class numbers, Math. Comp. 31 (1977), no. 138, 599–607. MR 432589, DOI 10.1090/S0025-5718-1977-0432589-6
- D. H. Lehmer and J. M. Masley, Table of the cyclotomic class numbers $h^*(p)$ and their factors for $200<p<521$, Math. Comp. 32 (1978), no. 142, 577–582. MR 498484, DOI 10.1090/S0025-5718-1978-0498484-2
- B. Mazur and A. Wiles, Class fields of abelian extensions of $\textbf {Q}$, Invent. Math. 76 (1984), no. 2, 179–330. MR 742853, DOI 10.1007/BF01388599
- Bernadette Perrin-Riou, Travaux de Kolyvagin et Rubin, Astérisque 189-190 (1990), Exp. No. 717, 69–106 (French). Séminaire Bourbaki, Vol. 1989/90. MR 1099872
- Karl Rubin, Kolyvagin’s system of Gauss sums, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 309–324. MR 1085265, DOI 10.1007/978-1-4612-0457-2_{1}4
- René Schoof, Cohomology of class groups of cyclotomic fields: an application to Morse-Smale diffeomorphisms, J. Pure Appl. Algebra 53 (1988), no. 1-2, 125–137. MR 955614, DOI 10.1016/0022-4049(88)90016-3
- René Schoof, The structure of the minus class groups of abelian number fields, Séminaire de Théorie des Nombres, Paris 1988–1989, Progr. Math., vol. 91, Birkhäuser Boston, Boston, MA, 1990, pp. 185–204. MR 1104706
- Schoof, R.: Class numbers of $\mathbb {Q} (\cos (2\pi /p))$, in preparation.
- David Solomon, On the classgroups of imaginary abelian fields, Ann. Inst. Fourier (Grenoble) 40 (1990), no. 3, 467–492 (English, with French summary). MR 1091830, DOI 10.5802/aif.1221
- F. J. van der Linden, Class number computations of real abelian number fields, Math. Comp. 39 (1982), no. 160, 693–707. MR 669662, DOI 10.1090/S0025-5718-1982-0669662-5
- Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674, DOI 10.1007/978-1-4684-0133-2
Additional Information
- René Schoof
- Affiliation: Dipartimento di Matematica, $2^{{a}}$ Università di Roma “Tor Vergata", I-00133 Rome, Italy
- Email: schoof@wins.uva.nl
- Received by editor(s): March 28, 1994
- Received by editor(s) in revised form: December 2, 1996
- © Copyright 1998 American Mathematical Society
- 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