The class number of $Q(\surd -p)$, for $P\equiv 1 (\textrm {mod}\ 8)$ a prime

Abstract:

Let $h( - p)$ be the class number of the quadratic field $Q(\surd - p)$, where $p \equiv 1\pmod 8$ is a prime. Write $p = {a^2} + {b^2} = 2{e^2} - {d^2}$, where $a \equiv e \equiv d \equiv b + 1 \equiv 1\pmod 2$ and $e > 0$. We prove that $h( - p) \equiv 0$ or $4\pmod 8$ according as $(e|p) = 1$ or $- 1$; using this, we prove that $h( - p) \equiv (p - 1)/2 + b\pmod 8$. The proofs are elementary, relying on the theory of composition of binary quadratic forms.