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

Konomi, Yutaka

doi:10.5802/jtnb.1101

On the

p

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

Konomi, Yutaka ¹

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

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

Abstract
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 $K$ .

Received: 2018-12-16
Revised: 2019-02-05
Accepted: 2019-02-18
Published online: 2020-05-06

DOI: 10.5802/jtnb.1101

Classification: 11R29, 20D06, 20D08
Keywords: Class group, simple group

Author's affiliations:

Konomi, Yutaka ¹

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

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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},
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     year = {2019},
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     url = {http://archive.numdam.org/articles/10.5802/jtnb.1101/}
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Konomi, Yutaka. On the $p$-rank of the ideal class group of a normal extension with simple Galois group. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 3, pp. 671-678. doi : 10.5802/jtnb.1101. http://archive.numdam.org/articles/10.5802/jtnb.1101/

References
Cited by

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

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

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

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