Note on the Kummer-Hilbert reciprocity law.
Katsumi Shiratani · Journal of the Mathematical Society of Japan · 1960
Let $p$ be an odd prime number, $Q$ the field of rational p-adic numbers, $\zeta$ a fixed primitive p-th root of unity, and $k=Q(\zeta)$ .The classical Kummer-Hilbert reciprocity law was purely locally proved by K. Yamamoto [8] in the following form.'Let $\mathfrak{p}$ be the prime ideal, and $\pi$ an arbitrary prime element in $k$ .By making use of the polynomialwe define differential quotients $l_{\pi}^{(i)}( u)$ , which are determined modulo $p$ , for a principal unit $ u$ in $k$ as follows:Then it is necessary and sufficient for $ u$ to be a norm of an element of $K=k(\sqrt[p]{\mu})$ , where $\mu$ is a principal unit in $k$ , that we have $\sum_{n=1}^{p-1}(-1)^{n-1}l_{\pi}^{(n)}( u)l_{\pi}^{(p-n)}(\mu)\equiv 0$ $(p)$ .