I'd like to know why if $K = \mathbb Q(\sqrt{-d})$, then $\mathcal O_K^* = \{\pm 1\}$ for $d \neq 1, 3$.
Dirichlet's unit theorem tells us that the only units in $\mathcal O_K$ are the roots of unity contained in $K$. Why does $\mathbb Q(\sqrt{-d})$ not contain any roots of unity other than $1,-1$ for the specified $d$?
Thanks