ON RELATIVE CLASS NUMBER AND CONTINUED FRACTIONS
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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.
Abstract
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.
참고문헌 (9)
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