On the ideal class groups of real abelian number fields
type: Item
instance of: scholarly article
title: On the ideal class groups of real abelian number fields
zbMATH Open document ID: 0665.12003
DOI: 10.2307/1971460
author: Francisco Thaine
published in: Annals of Mathematics. Second Series
publication date: 1988-01-01T00:00:00Z
review text: This paper presents a new relationship between the ideal class group \(A\) and the unit group \(E\) of a real abelian field \(K\). In fact, the author finds annihilators of the \(p\)-class group \((A)_ p\) related to the structure of \(W=E/C\) (\(C=\) circular units), and in this way obtains smooth statements about relations between annihilators of \((A)_ p\) and \((W)_ p\). Let \(m\) denote the conductor of \(K\), let \(\zeta_ n\) be a primitive \(n\)th root of unity, and put \(\Delta =\text{Gal}(K/\mathbb Q)\). Define the group \(C\) of circular units of \(K\) to consist of all units of the form \(f(1)\), where \[ f(X)=\pm \prod^{j}_{i=1}\prod^{m-1}_{k=1}(X^ i-\zeta^ k_ m)^{a_{ik}} \in K(X) \] with \(a_{ik}\in\mathbb Z\) and \(j\geq 1\) (the author points out that this definition differs from the known and may produce a larger group). Consider first a field \(L=K(\zeta_ q)\), where \(q\) is a sufficiently large prime that splits completely in \(K\). Starting with a unit \(\delta =f(1)\in C\) the author observes that a generating automorphism of \(L/K\) maps certain elements \(\alpha\in L\) to \(f(\zeta_ q)\alpha\). On studying the prime factorization of \(\alpha\) he infers, back to \(K\), a decomposition \[ (N_{L/K} (\alpha)) = R^ b \prod_{\sigma\in\Delta} \sigma^{-1} (Q)^{r_\sigma}, \tag{\(*\)} \] valid under certain conditions, where \(R\) and \(Q\) are ideals of \(K\), \(Q\) a prime above \(q\), and \(b\) is a divisor of \(q-1\). The exponents \(r_{\sigma}\) have a deep connection to the units of \(K\). Here a crucial role is played by a local-global theorem depending on Chebotarev's density theorem. From (*) it follows that if \(R^ b\) is principal then \(\sum_{\sigma \in \Delta}r_{\sigma}\sigma^{-1} \) annihilates the ideal class \({\mathcal C}\) containing \(Q\). By use of the properties of \(r_{\sigma}\) this implies the following results: If \(b=p^ n\) is the exponent of \((A)_ p\) and if \(\delta\) satisfies, for all \(\sigma\in \Delta\), a congruence \(…
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11G16
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 4087779
zbMATH Keywords: ideal class group
zbMATH Keywords: unit group
zbMATH Keywords: real abelian field
zbMATH Keywords: annihilators of the p-class group
zbMATH Keywords: circular units
zbMATH Keywords: local-global theorem
zbMATH Keywords: density
MaRDI profile type: Publication
Wikidata QID: Q56454674
full work available at URL: https://doi.org/10.2307/1971460
OpenAlex ID: W2013337587
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description: scientific article; zbMATH DE number 4087779