$ℚ (\sqrt - 3)$

Consider the monogenic imaginary quadratic field $𝐾 = ℚ (𝛼)$ where $𝛼 = \sqrt - 3$ . alg

Sage

K.<α> = QuadraticField(-3)

Discriminant

Δ 𝐾 = - 3 .

By ^P1, we have the exceptional case

O \times 𝐾 = {\pm 1, \pm 1 - 𝛼 2, \pm 1 + 𝛼 2} ≅ C 6

𝑀 𝐾 = 2 \sqrt 3 𝜋 < 2

hence $C l 𝐾$ is trivial, i.e. $O 𝐾$ is a UFD.