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ON THE CLASS NUMBER AND THE FUNDAMENTAL UNIT OF THE REAL QUADRATIC FIELD
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 85 / Issue 3 / June 2012
- Published online by Cambridge University Press:
- 25 January 2012, pp. 359-370
- Print publication:
- June 2012
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- Article
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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.
, let 