## Journal of Commutative Algebra

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

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

Yutaka Konomi and Takayuki Morisawa

#### 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