Variations on the theme of D. K. Faddeev's paper ``An explicit form of the Kummer-Takagi reciprocity law''

The following form of the Eisenstein reciprocity law is established: in the cyclotomic field $\mathbb{Q}(\zeta)$, the relation $(\frac{\alpha}{a})=(\frac{a}{\alpha})$ is equivalent to $\frac{a^{p-1}-1}{p}\cdot \underline{\alpha}'(1)\equiv 0\mod p$.