Is the following proposition true? If yes, how would you prove this?
Proposition Let $f(X) = X^3 + aX + b$ be an irreducible polynomial in $\mathbb{Z}[X]$. Let $d = -(4a^3 + 27b^2)$ be the discriminant of $f(X)$. Let $K = \mathbb{Q}(\sqrt{d})$. Let $L$ be the splitting field of $f(X)$ over $\mathbb{Q}$. Suppose the following conditions hold.
(1) $|d| = |4a^3 + 27b^2|$ is a prime number.
(2) The class number of $K$ is 3.
(3) $f(X) \equiv (X - s)^2(X - t)$ (mod $d$), where $s$ and $t$ are distinct rational integers mod $d$.
Then $L$ is the Hilbert class field of $K$.
Examples Each of the following polynomials of negative discriminants satisfies the above conditions.
(1) $f(X) = X^3 - X + 1 \equiv (X - 13)^2(X - 20)$ (mod 23)
(2) $f(X) = X^3 + X + 1 \equiv (X-3)(X-14)^2$ (mod 31)
(3) $f(X) = X^3 + 2X + 1 \equiv (X - 14)^2(X - 31)$ (mod 59)
I could not find a polynomial of positive discriminant satisying the above conditions.