Sign of image of square root under automorphism of cyclotomic extension

Question

Suppose $m \ne 0, 1$ is a square free integer, $K = \mathbb{Q}(\sqrt m)$ and the ”discriminant” $∆=m$ if $d≡1(mod4)$, and $∆=4m$ otherwise. I am looking for a reference of the fact that the Kronecker symbol $(\frac{m}{r})$ is $\pm 1$ depending on whether the automorphism $\zeta → \zeta^r$ fixes $\sqrt d$ or not. I interpret the statement as $$ \sigma_r\left(\sqrt m\right)=\left(\frac{m}{r}\right)\sqrt m,$$ where $(r, \Delta m)=1$ and $\sigma_r \in Aut(\mathbb{Q}(\zeta_{\Delta m}))$. Is this correct ?