Prime factors of class number of cyclotomic fields
Journal de Théorie des Nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 525-530.

Let p be an odd prime, r be a primitive root modulo p and r i r i (modp) with 1r i p-1. In 2007, R. Queme raised the question whether the -rank ( an odd prime p) of the ideal class group of the p-th cyclotomic field is equal to the degree of the greatest common divisor over the finite field 𝔽 of x (p-1)/2 +1 and Kummer’s polynomial f(x)= i=0 p-2 r -i x i . In this paper, we shall give the complete answer for this question enumerating a counter-example.

Soit p un nombre premier impair, r une racine primitive modulo p et r i r i (modp) avec 1r i p-1. En 2007, R. Queme a posé la question : le -rang ( premier impair p) du groupe des classes d’idéaux du p-ième corps cyclotomique est-il égal au degré du plus grand diviseur commun sur le corps fini 𝔽 de x (p-1)/2 +1 et du polynôme de Kummer f(x)= i=0 p-2 r -i x i . Dans cet article, nous donnons une réponse complète à cette question en produisant un contre-exemple.

Received:
Published online:
DOI: 10.5802/jtnb.639
Tetsuya Taniguchi 1

1 Department of Mathematics, Tokyo University of Science, Noda, Chiba 278-8510, Japan
@article{JTNB_2008__20_2_525_0,
     author = {Tetsuya Taniguchi},
     title = {Prime factors of class number of cyclotomic fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {525--530},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {2},
     year = {2008},
     doi = {10.5802/jtnb.639},
     zbl = {1163.11078},
     mrnumber = {2477516},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.639/}
}
TY  - JOUR
TI  - Prime factors of class number of cyclotomic fields
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2008
DA  - 2008///
SP  - 525
EP  - 530
VL  - 20
IS  - 2
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.639/
UR  - https://zbmath.org/?q=an%3A1163.11078
UR  - https://www.ams.org/mathscinet-getitem?mr=2477516
UR  - https://doi.org/10.5802/jtnb.639
DO  - 10.5802/jtnb.639
LA  - en
ID  - JTNB_2008__20_2_525_0
ER  - 
%0 Journal Article
%T Prime factors of class number of cyclotomic fields
%J Journal de Théorie des Nombres de Bordeaux
%D 2008
%P 525-530
%V 20
%N 2
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.639
%R 10.5802/jtnb.639
%G en
%F JTNB_2008__20_2_525_0
Tetsuya Taniguchi. Prime factors of class number of cyclotomic fields. Journal de Théorie des Nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 525-530. doi : 10.5802/jtnb.639. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.639/

[1] Tom M. Apostol Resultants of cyclotomic polynomials, Proc. Amer. Math. Soc., Volume 24 (1970), pp. 457-462 | MR: 251010 | Zbl: 0188.34002

[2] H. Kisilevsky Olga Taussky-Todd’s work in class field theory, Pacific J. Math. (1997) no. Special Issue, pp. 219-224 | Zbl: 1011.11002

[3] Eduard Ernst Kummer Bestimmung der Anzahl nicht äquivalenter Classen für die aus λ ten Wurzeln der Einheit gebildeten complexen Zahlen und die idealen Factoren derselben, J. Reine Angew. Math., Volume 40 (1850), pp. 43-116 | Zbl: 040.1088cj

[4] D. H. Lehmer Prime factors of cyclotomic class numbers, Math. Comp., Volume 31 (1977), pp. 599-607 | MR: 432589 | Zbl: 0357.12006

[5] M. Pohst; H. Zassenhaus Algorithmic algebraic number theory, Encyclopedia of Mathematics and its Applications, Volume 30, Cambridge Univ. Press, Cambridge, 1997 | MR: 1483321 | Zbl: 0685.12001

[6] René Schoof Minus class groups of the fields of the lth roots of unity, Math. Comp., Volume 67 (1998), pp. 1225-1245 | MR: 1458225 | Zbl: 0902.11043

[7] Tetsuya Taniguchi Program codes of “Prime factors of class number of cyclotomic fields” (http://www.ma.noda.tus.ac.jp/g/tt/jtnb2008/)

[8] Lawrence C. Washington Introduction to cyclotomic fields, Springer-Verlag, New York, 2nd ed., 1997 | MR: 1421575 | Zbl: 0966.11047

Cited by Sources: