Let $K$ be a finite extension of $\mathbb Q_p$ and $L/K$ be a finite Galois extension. Then also $L(T)/K(T)$ and $L((T))/K((T))$ are Galois extensions with Galois group isomorphic to ${\rm Gal}(L/K)$ by having this group act on coefficients (make it act on $T$ trivially).
Let $N: L((T)) \to K((T))$ denote the norm map. It restricts to norm maps $L(T) \to K(T)$, $L[[T]] \to K[[T]]$, $L[T] \to K[T]$, and $L \to K$. I am interested in converses when the norm value is a rational function of polynomial: for $f(T) \in L((T))$, if $N(f(T)) \in K(T)$ then is $f(T) \in L(T)$? If $N(f(T)) \in K[T]$ then is $f(T) \in L[T]$? (One can ask similar questions with $L$ and $K$ replaced by their rings of integers $O_L$ and $O_K$.) If the answer to these two questions is no, does it become yes if we assume the Galois group is cyclic or is a $p$-group (or both)?
I have tried taking a generic power series and computing its norm in terms of its coefficients. The coefficients of the norm are sums of $d$-term products of Galois twists of the original coefficients, where $d = [L:K]$. So when the norm is a polynomial, coefficients with large enough index must vanish, which gives us equations for the original coefficients. However, it is not clear whether this implies what we want.