# 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.