Computing the Class Number H(-3)

Theorem 1: The class number $H(-3) = 1$.

• Proof: Let $f(x, y)$ be a reduced binary quadratic form. Since $d = -3 < 0$, we must have that:

(1)

\begin{align} \quad 0 < a \leq \sqrt{\frac{-d}{3}} = \sqrt{1} = 1 \end{align}

• So $a = 1$. But then $0 < b \leq a = c$ implies that $c = 1$ and $b = 1$, so $f(x, y) = x^2 + xy + y^2$ is a reduced binary quadratic form with discriminant $d = -3$. The inequality $-1 < b \leq a < c$ implies that $b = 0$ or $b = 1$. If $b = 0$ then $-3 = b^2 - 4ac = -4c$ can never be satisfied. If $b = 1$ then $-3 = b^2 - 4ac = 1 - 4c$ implies that $-4 = 4c$, so $c = 1$, which gives us $f(x, y)$.

• So the only reduced binary quadratic form with discriminant $d = -3$ is $f(x, y)$ and $H(-3) = 1$. $\blacksquare$