Let $L/K$ be a finite cyclic extension of number fields, let $\mathfrak{p}$ be a prime of $K$, and let $\mathfrak{P}$ be a prime of $L$ lying above $K$. If $x \in N_{L/K}(L^*) \subset K^*$, why is it true that $x \in N_{L_\mathfrak{P}/K_\mathfrak{p}}(L_{\mathfrak{P}}^*)$? I.e., why are global norms also local norms at every place?
I know that $$N_{L/K}(y) = \prod_{\mathfrak{P} \mid \mathfrak{p}} N_{L_\mathfrak{P}/K_\mathfrak{p}}(y),$$ but I don't know exactly how or whether to use this.
In your product formula, note that the $\frak P$ are conjugate by the Galois group $G$ of $L/K$, so that if we pick any one of them, say $\frak P_0$ then each one has the form $\sigma({\frak P_0})$ for some $\sigma\in G$. Then $$N_{\frak P/p}(y)=N_{\frak \sigma(P_0)/p}(y)= N_{\frak P_0/p}(\sigma^{-1}(y)).$$ Therefore each term in your product is an element of the group $N_{\frak P_0/p}(L_{\frak P_0}^*)$ and so the whole product is.