Class number of $\mathbb{Q}(\sqrt{n})$ always even? Let $n$ be a negative square-free even integer. Does it necessarily follow that the class number of $\mathbb{Q}(\sqrt{n})$ is even?

Gauss proved that the class number $h(d)$ of $\mathbb{Q}(\sqrt{d})$, for squarefree $d<0$ is divisible by $2^{s-1}$ for $d\equiv 1,2 \bmod 4$, and divisible by $2^s$ for $d\equiv 3\bmod 4$, see here. Here $s$ denotes the number of prime factors of $d$. If $d\neq -2$ is even, $s\ge 2$ and we are done. So the answer is: except for $\mathbb{Q}(\sqrt{-2})$, which has class number $1$, all other 
imaginary quadratic number fields  $\mathbb{Q}(\sqrt{-n})$, for even $n \neq -2$ have even class numer.
