Journal of Commutative Algebra

On the class semigroup of the cyclotomic $\mathbb Z_p$-extension of the rational numbers

Yutaka Konomi and Takayuki Morisawa

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Abstract

For a commutative integral domain, the class semigroup and the class group are defined as the quotient of the semigroup of fractional ideals and the group of invertible ideals by the group of principal ideals, respectively. Let $p$ be a prime number. In algebraic number theory, especially in Iwasawa theory, the class group of the ring of integers $\mathcal{O} $ of the cyclotomic $\mathbb {Z}_{p}$-extension of the rational numbers has been studied for a long time. However, the class semigroup of $\mathcal{O} $ is not well known. We are interested in the structure of the class semigroup of $\mathcal{O} $. In order to study it, we focus on the structure of the complement set of the class group in the class semigroup of $\mathcal{O} $. In this paper, we prove that the complement set is a group and determine its structure.

Article information

Source
J. Commut. Algebra, Volume 11, Number 1 (2019), 69-80.

Dates
First available in Project Euclid: 13 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.jca/1552464133

Digital Object Identifier
doi:10.1216/JCA-2019-11-1-69

Mathematical Reviews number (MathSciNet)
MR3922426

Zentralblatt MATH identifier
07037589

Subjects
Primary: 11R23: Iwasawa theory 11R29: Class numbers, class groups, discriminants

Keywords
Class semigroup class group $\mathbb Z_p$-extension

Citation

Konomi, Yutaka; Morisawa, Takayuki. On the class semigroup of the cyclotomic $\mathbb Z_p$-extension of the rational numbers. J. Commut. Algebra 11 (2019), no. 1, 69--80. doi:10.1216/JCA-2019-11-1-69. https://projecteuclid.org/euclid.jca/1552464133


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