On the $p$-rank of the ideal class group of a normal extension with simple Galois group

Yutaka Konomi

doi:10.5802/jtnb.1101

On the

p

-rank of the ideal class group of a normal extension with simple Galois group

Yutaka Konomi ¹

¹ Department of Mathematics Meijo University Tempaku-ku, Nagoya 468-8502, Japan

Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 671-678.

Abstract (VO)
Résumé

Let $p$ be a prime and $L$ a finite normal extension over a number field $K$ whose Galois group is simple and non-abelian. The aim of this paper is to estimate a lower bound of the ratio of the $p$ -rank of the ideal class group of $L$ to the $p$ -rank of the ambiguous $p$ -class group of $L$ with respect to $K$ .

Soient $p$ un nombre premier et $L$ une extension normale finie d’un corps de nombres $K$ dont le groupe de Galois est simple et non abélien. Le but de cet article est d’estimer la borne inférieure du quotient du $p$ -rang du groupe de classes d’idéaux de $L$ par le $p$ -rang du groupe de $p$ -classes ambiges de $L$ par rapport à $K$ .

Reçu le : 2018-12-15
Révisé le : 2019-02-04
Accepté le : 2019-02-17
Publié le : 2020-05-06

MR Numdam Zbl

DOI : 10.5802/jtnb.1101

Classification : 11R29, 20D06, 20D08
Mots-clés : Class group, simple group

Affiliations des auteurs :

Yutaka Konomi ¹

¹ Department of Mathematics Meijo University Tempaku-ku, Nagoya 468-8502, Japan

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On the $p$-rank of the ideal class group of a normal extension with simple {Galois} group},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {671--678},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {3},
     year = {2019},
     doi = {10.5802/jtnb.1101},
     zbl = {1426.11126},
     mrnumber = {4102621},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1101/}
}

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Yutaka Konomi. On the $p$-rank of the ideal class group of a normal extension with simple Galois group. Journal de théorie des nombres de Bordeaux, Tome 31 (2019) no. 3, pp. 671-678. doi : 10.5802/jtnb.1101. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1101/

Bibliographie
Cité par

[1] J. T. Goozeff Abelian $p$ -subgroups of the general linear group, J. Aust. Math. Soc., Volume 11 (1970), pp. 257-259 | DOI | MR | Zbl

[2] Yutaka Konomi On $p$ -class group of an $A_{n}$ -extension, Proc. Japan Acad., Volume 84 (2008) no. 7, pp. 87-88 | DOI | MR | Zbl

[3] G. N. Thwaites The Abelian $p$ -subgroups of $GL (n, p)$ of maximal rank, Bull. Lond. Math. Soc., Volume 4 (1972), pp. 313-320 | DOI | MR | Zbl

Yutaka Konomi On the ideal class group on which the alternating group of degree five acts and special values of Artin L-functions, The Ramanujan Journal, Volume 59 (2022) no. 1, p. 87 | DOI:10.1007/s11139-021-00540-6
Tatsuya Ohshita On infinite extensions of Dedekind domains, upper semicontinuous functions and the ideal class semigroups, Journal of Commutative Algebra, Volume 13 (2021) no. 3, pp. 407-434 | DOI:10.1216/jca.2021.13.407 | Zbl:1483.11248
Yutaka Konomi; Takayuki Morisawa On the class semigroup of $Z_{S}$ -fields and Iwasawa invariants, Journal of Pure and Applied Algebra, Volume 223 (2019) no. 9, pp. 3665-3680 | DOI:10.1016/j.jpaa.2018.12.001 | Zbl:1470.11282

Cité par 3 documents. Sources : Crossref, zbMATH