Quadratic number fields with class numbers divisible by a prime q

Let q≥5 be a prime number. Let
$k={\BBQ}(\sqrt{d})$
be a quadratic number field, where
$d = (-1)^{(q(q-1))/(2)}\bullet (-(q-1)^{q-1}uw^{q}+u^{2}q^{q})$
. Then the class number of k is divisible by q for certain integers u, w. Conversely, assume Ω/k is an unramified cyclic extension of degree q(which implies the class number of k is divisible by q), and Ω is the splitting field of some irreducible trinomial f(X) = X
^q
-aX-b with integer coefficients,
$k={\BBQ}(\sqrt{D(f)})$
with D(f) the discriminant of f(X). Then f(X) must be of the form f(X= X
^q
-u
^q−2
wX-u
^q−1
in a certain sense where u, w are certain integers. Therefore,
$k={\BBQ}(\sqrt{d})$
with
$d = (-1)^{(q(q-1))/(2)} (-(q-1)^{q-1}uw^{q}+u^{2}q^{q})$
. Moreover, the above two results are both generalized for certain kinds of general polynomials.