Ideal class groups of real quadratic fields of generalized ERD-type

Let
$K = Q(\bar{d})$
and d = s
²
+ r > 0. If r| 4s then K is the ERD-type. Here it is generalized to the types with r| 2
ⁿ
s, r| 4z
ⁿ
s, or r| z
ⁿ
s. Special properties of the class number h(d) and class group H(d) of K are proved. In particular, if d = a
²
t
²
± 2
^q+2
t with prime q, then q| h(d) and H(d) contains a cyclic subgroup of order q provided 4 and 16 are not complete denominator of the simple continued-fraction of
$\bar{d}$
. Both d = a
²
t
²
± 4z
^q
t and d = a
²
t
²
/4 ± z
^q
t have similar specialties.