Prime factors of class number of cyclotomic fields

Tetsuya Taniguchi[1]

  • [1] Department of Mathematics, Tokyo University of Science, Noda, Chiba 278-8510, Japan

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 2, page 525-530
  • ISSN: 1246-7405

Abstract

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Let p be an odd prime, r be a primitive root modulo p and r i ≡ r i ( mod p ) with 1 ≤ r i ≤ p - 1 . In 2007, R. Queme raised the question whether the ℓ -rank ( ℓ 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 ℓ 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.

How to cite

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Taniguchi, Tetsuya. "Prime factors of class number of cyclotomic fields." Journal de Théorie des Nombres de Bordeaux 20.2 (2008): 525-530. <http://eudml.org/doc/10848>.

@article{Taniguchi2008,
abstract = {Let $p$ be an odd prime, $r$ be a primitive root modulo $p$ and $r_\{i\} \equiv r^\{i\} \hspace\{4.44443pt\}(\@mod \; p)$ with $1 \le r_i \le p-1$. In 2007, R. Queme raised the question whether the $\ell $-rank ($\ell $ an odd prime $ \ne 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 $\mathbb\{F\}_\ell $ of $x^\{(p-1)/2\}+1$ and Kummer’s polynomial $f(x) = \sum _\{i=0\}^\{p-2\} r_\{-i\} x^i$. In this paper, we shall give the complete answer for this question enumerating a counter-example.},
affiliation = {Department of Mathematics, Tokyo University of Science, Noda, Chiba 278-8510, Japan},
author = {Taniguchi, Tetsuya},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {cyclotomic field; minus class number; rank; class group},
language = {eng},
number = {2},
pages = {525-530},
publisher = {Université Bordeaux 1},
title = {Prime factors of class number of cyclotomic fields},
url = {http://eudml.org/doc/10848},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Taniguchi, Tetsuya
TI - Prime factors of class number of cyclotomic fields
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 2
SP - 525
EP - 530
AB - Let $p$ be an odd prime, $r$ be a primitive root modulo $p$ and $r_{i} \equiv r^{i} \hspace{4.44443pt}(\@mod \; p)$ with $1 \le r_i \le p-1$. In 2007, R. Queme raised the question whether the $\ell $-rank ($\ell $ an odd prime $ \ne 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 $\mathbb{F}_\ell $ of $x^{(p-1)/2}+1$ and Kummer’s polynomial $f(x) = \sum _{i=0}^{p-2} r_{-i} x^i$. In this paper, we shall give the complete answer for this question enumerating a counter-example.
LA - eng
KW - cyclotomic field; minus class number; rank; class group
UR - http://eudml.org/doc/10848
ER -

References

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  1. Tom M. Apostol, Resultants of cyclotomic polynomials, Proc. Amer. Math. Soc. 24 (1970), 457-462 Zbl0188.34002MR251010
  2. H. Kisilevsky, Olga Taussky-Todd’s work in class field theory, Pacific J. Math. (1997), 219-224 Zbl1011.11002
  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. 40 (1850), 43-116 
  4. D. H. Lehmer, Prime factors of cyclotomic class numbers, Math. Comp. 31 (1977), 599-607 Zbl0357.12006MR432589
  5. M. Pohst, H. Zassenhaus, Algorithmic algebraic number theory, 30 (1997), Cambridge Univ. Press, Cambridge Zbl0685.12001MR1483321
  6. René Schoof, Minus class groups of the fields of the l th roots of unity, Math. Comp. 67 (1998), 1225-1245 Zbl0902.11043MR1458225
  7. Tetsuya Taniguchi, Program codes of “Prime factors of class number of cyclotomic fields” Zbl1163.11078
  8. Lawrence C. Washington, Introduction to cyclotomic fields, (1997), Springer-Verlag, New York, 2nd ed. Zbl0966.11047MR1421575

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