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New invariants and class number problem in real quadratic fields

Published online by Cambridge University Press:  22 January 2016

Hideo Yokoi
Hideo Yokoi*
Affiliation:
Graduate School of Human Informatics Nagoya University Chikusa-ku, Nagoya 464-01Japan
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In recent papers [10, 11, 12, 13, 14], we defined some new ρ-invariants for any rational prime ρ congruent to 1 mod 4 and D-invariants for any positive square-free integer D such that the fundamental unit εD of real quadratic field Q(√D) satisfies NεD = –1, and studied relationships among these new invariants and already known invariants.

One of our main purposes in this paper is to generalize these D-invariants to invariants valid for all square-free positive integers containing D with NεD = 1. Another is to provide an improvement of the theorem in [14] related closely to class number one problem of real quadratic fields. Namely, we provide, in a sense, a most appreciable estimation of the fundamental unit to be able to apply, as usual (cf. [3, 4, 5, 9, 12, 13]), Tatuzawa’s lower bound of L(l, XD) (Cf[7]) for estimating the class number of Q(√D) from below by using Dirichlet’s classical class number formula.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 132 , December 1993 , pp. 175 - 197
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1993

References

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