Class Number 1 Criteria for Real Quadratic Fields of Richaud–Degert Type

Class Number 2 Criteria for Real Quadratic Fields of Richaud–Degert Type

“…This is one of the main theorems of [2]. The basic idea of the proof is the same as that of Theorem 2.5 below.…”

“…(ii) 2 ramifies in k if d#2, 3 (mod 4), i.e., (2)=(2, :+-d ) 2 where :=0 if d#2 (mod 4) and :=1 if d#3 (mod 4).…”

A note on class number 1 criteria for totally real algebraic number fields

“…For such a field, the fact that the condition in Corollary 3.2 is also necessary for the class number of K to be 1 was proved by many people with different methods. For examples, see [1], [5], [6], [11].…”

Class Number One Criterion for Some Non-Normal Totally Real Cubic Fields

“…, q − 2. For real quadratic fields, many authors [2,3,8,11] considered the connection between prime producing polynomials and class number. For the simplest cubic fields, Kim and Hwang [6] gave a class number one criterion which is related to some prime producing polynomials.…”