Some property of norm residue symbol

Question

On the last page of this article, Artin and Hasse used some property of the norm residue symbol $$\left(\frac{\lambda}{A}\right)_K = \left(\frac{\lambda}{n(A)}\right)_{k_\zeta}$$ (under Beweis von Satz 3) but I cannot seem to find any reference for this or any easy explanation. In the context of the paper, $k_\zeta = \mathbb{Q}_\ell(\zeta)$ where $\zeta$ is a primitive $\ell^{\text{th}}$ root of unity with $\lambda = 1 - \zeta$ its uniformizer, $K$ is any field extension of $k_\zeta$, $A \equiv 1 \bmod \lambda$ in $K$, and $n$ denotes the relative norm $K/k_\zeta$. Is there an easy explanation and/or some generalization?