Real Quadratic Fields In Which Every Non-Maximal Order Has Relative Class Number Greater Than One

TL;DR

This paper investigates the relative class numbers of non-maximal orders in real quadratic fields, proving that for certain fields, no non-maximal order has a relative class number equal to one, thus addressing a question posed by Cohn.

Contribution

The paper demonstrates that in specific real quadratic fields, such as when m=46, no non-maximal order has a relative class number of one, providing new insights into class number behavior.

Findings

No non-maximal order has relative class number one for m=46

Confirmed similar results in seven other cases

Addresses Cohn's question on class numbers in quadratic fields

Abstract

Cohn asks if for every real quadratic field Q(m) with discriminant d there exists a non-maximal order corresponding to f > 1 such that the relative class number Hd(f) = h(f2d)/h(d) is one. We prove that when m = 46 (and in seven other cases) there is no such order.