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ON THE CLASS NUMBER AND THE FUNDAMENTAL UNIT OF THE REAL QUADRATIC FIELD

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  25 January 2012

JAE MOON KIM  and
JADO RYU
JAE MOON KIM
Affiliation:
Department of Mathematics, Inha University, Incheon 402-751, Korea (email: jmkim@inha.ac.kr)
JADO RYU*
Affiliation:
Department of Mathematics, Inha University, Incheon 402-751, Korea (email: jdryu@inha.ac.kr)
*
✉For correspondence; e-mail: namver@naver.com
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Abstract

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For a real quadratic field , let tk be the exact power of 2 dividing the class number hk of k and ηk the fundamental unit of k. The aim of this paper is to study tk and the value of Nk/ℚ(ηk). Various methods have been successfully applied to obtain results related to this topic. The idea of our work is to select a special circular unit ℰk of k and investigate C(k)=〈±ℰk 〉. We examine the indices [E(k):C(k)] and [C(k):CS (k)] , where E(k) is the group of units of k, and CS (k) is that of circular units of k defined by Sinnott. Then by using the Sinnott’s index formula [E(k):CS (k)]=hk, we obtain as much information about tk and Nk/ℚ (ηk) as possible.

Keywords

class numberfundamental unitcircular unitunit indexindex formula

MSC classification

Secondary: 11R11: Quadratic extensions
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 85 , Issue 3 , June 2012 , pp. 359 - 370
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

This work was supported by INHA UNIVERSITY Research Grant.

References

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