Let $K/k$ be cyclic, and $\sigma$ a generator of $G(K/k)$. Let $F$ be an extesion of $k$ with $k\subset F\subset K$ and $[K:F] = m$. Let $a\neq 0 \in k,$ and assume that $a^m = N_{K/k}(b)$ for some $b\in K$. Show that $\exists c\in F$ such that $N_{F/k}(c)= a$.
(This is actually a question from P.J. McCarthy's ''Algebraic Extensions of Fields,'' Ch.2 Exercise 31. )