Variations on the theme of D. K. Faddeev’s paper “An explicit form of the Kummer–Takagi reciprocity law”

Abstract:

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 $\frac {a^{p-1}-1}{p}\cdot \underline {\alpha }’(1)\equiv 0\mod p$.