A note on quadratic fields in which a fixed prime number splits completetly. III
type: Item
instance of: scholarly article
title: A note on quadratic fields in which a fixed prime number splits completetly. III
zbMATH Open document ID: 0963.11062
author: Humio Ichimura
published in: Proceedings of the Japan Academy. Series A
publication date: 2001-04-03T00:00:00Z
review text: For a fixed prime number \(p\), denote by \(M(p)^+\) the set of all real quadratic fields in which \(p\) splits, and for a quadratic field \(K\) in \(M(p)^+\) denote by \(\delta_p^+(K)\) the order of the ideal class of \(K\) containing a prime ideal of \(K\) over \(p\). Then the author is concerned with the image of the map \(\delta_p^+: M(p)^+\to N\), \(K\to \delta_p^+(K)\). In Part II of his series [Proc. Japan Acad., Ser. A 75, 150-151 (1999; Zbl 0997.11088)], he showed that the image of \(\delta_p^+\) contains \(2n\) for all \(n\geq 0\) and any \(p\). In this paper, he shows the following: Under the assumption of the \(abc\) conjecture [cf. \textit{P. Vojta}, Diophantine approximations and value distribution theory, Chap. 5, Lecture Notes Math. 1239, Springer (1987; Zbl 0609.14011)], (i) the complement \(N\setminus \operatorname {Im} \delta_p^+\) is a finite set for any prime \(p\), (ii) \(\operatorname {Im} \delta_p^+\) coincides with \(N\) for infinitely many \(p\).
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1443394
zbMATH Keywords: real quadratic fields
zbMATH Keywords: ideal class
zbMATH Keywords: \(abc\) conjecture
reviewed by: Hideo Yokoi
MaRDI profile type: Publication
full work available at URL: pjaa.75.176
OpenAlex ID: W2050676767
cites work: On the class-number of the maximal real subfield of a cyclotomic field.
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cites work: A note on quadratic fields in which a fixed prime number splits completely
cites work: A note on quadratic fields in which a fixed prime number splits completely. II
cites work: On the Insolubility of a Class of Diophantine Equations and the Nontriviality of the Class Numbers of Related Real Quadratic Fields of Richaud-Degert Type
cites work: Diophantine approximations and value distribution theory
cites work: Real quadratic fields with class numbers divisible by \(n\)
cites work: Q3815362
DOI: 10.3792/PJAA.75.176
description: scientific article; zbMATH DE number 1443394