I am currently trying to prove the following from Washington's book on cyclotomic fields:

Let $q\equiv 3 \pmod{4}$ be prime, such that $R$ and $N$ denote the number of quadratic residues and non-residues modulo $q$ respectively in the interval $\left[1,\frac{q-1}{2}\right]$. Use Stickelberger's theorem to show that $R-N$ annihilates the class group of $\mathbb{Q}(\sqrt{-q})$.

Taking the obvious first step, as $\mathbb{Q}(\zeta_q)\supset\mathbb{Q}(\sqrt{-q})$ is the smallest cyclotomic field to contain $\mathbb{Q}(\sqrt{-q})$ we can calculate the Stickelberger element $\theta(\mathbb{Q}(\sqrt{-q}))$. A fairly easy calculation gives that:

$$\Theta=\Theta(\mathbb{Q}(\sqrt{-q})) = \frac{q^2-1}{4q}+\frac{(q-1)^2}{4q}\sigma$$

My question is then how do I take this and show that $R-N \in \mathbb{Z}\cap\Theta\mathbb{Z}$, as I presume is required to show that it annihilates the class group?


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