Let $K,L,M$ and $T$ be a fields. Suppose that $K/L$ and $T/M$ are galois extensions, and define $G=Gal(K/L)$ and $H=Gal(T/M)$ with $G\cong H$.
Consider $\alpha \in K$ such that $f(\alpha) \in T$ for some $f \in M[x]$.
Is $N_{T/M}(f(\alpha))=f(N_{K/L}(\alpha))?$