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2019 On the class semigroup of the cyclotomic $\mathbb Z_p$-extension of the rational numbers
Yutaka Konomi, Takayuki Morisawa
J. Commut. Algebra 11(1): 69-80 (2019). DOI: 10.1216/JCA-2019-11-1-69

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.

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Yutaka Konomi. Takayuki Morisawa. "On the class semigroup of the cyclotomic $\mathbb Z_p$-extension of the rational numbers." J. Commut. Algebra 11 (1) 69 - 80, 2019. https://doi.org/10.1216/JCA-2019-11-1-69

Information

Published: 2019
First available in Project Euclid: 13 March 2019

zbMATH: 07037589
MathSciNet: MR3922426
Digital Object Identifier: 10.1216/JCA-2019-11-1-69

Subjects:
Primary: 11R23, 11R29

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

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Vol.11 • No. 1 • 2019
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