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The relative class number Hd(f) of a real quadratic field K = Q(√m) of discriminant d is the ratio of class numbers of Of and OK, where OK denotes the ring of integers of K and Of is the order of conductor f given by Z+fOK. In a recent paper of A. Furness and E. A.
Parker the relative class number of Q(√m) has been investigated using continued fraction in the special case when √m has a diagonal form.
Here, we extend their result and show that there exists a conductor f of relative class number 1 when the continued fraction of √m is non- diagonal of period 4 or 5. We also show that there exist infinitely many real quadratic fields with any power of 2 as relative class number if there are infinitely many Mersenne primes.
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참고문헌 (9)
참고문헌 신청
A. Furness / 2012 / On Dirichlet’s conjecture on relative class number one / J. Number Theory 132 (7) : 1398 ~ 1403
D. Chakraborty / 2014 / Another look at real quadratic fields of relative class number 1 / Acta Arith 163 (4) : 371 ~ 378
H. Cohn / 1962 / A numerical study of the relative class numbers of real quadratic integral do-mains / Math. Comp 16 : 127 ~ 140
H. Davenport / 2008 / Higher Arithmetic / Cambridge University Press
I. Niven / 2008 / An Introduction to the Theory of Numbers / John Wiley and Sons Inc
D. Chakraborty / 2014 / Another look at real quadratic fields of relative class number 1 / Acta Arith 163 (4) : 371 ~ 378
H. Cohn / 1962 / A numerical study of the relative class numbers of real quadratic integral do-mains / Math. Comp 16 : 127 ~ 140
H. Davenport / 2008 / Higher Arithmetic / Cambridge University Press
I. Niven / 2008 / An Introduction to the Theory of Numbers / John Wiley and Sons Inc
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