nt.number theory – Imaginary quadratic fields with $ell$-indivisible class number

Let $ell geq 5$ be a prime. Show that there exists an imaginary quadratic field $K$ with odd fundamental discriminant:

(a) $ell$ inert in $K$,
(b) $(ell, h_{K})=1$.

Remark. Without requiring the discriminant being odd, the existence due to Horie-Onishi.