Suppose $K$ is a local field and $L$ a finite cyclic extension of $K$. By Hilbert 90, we know that if an element $a$ in $ L$ such that $N_{L / K}(a) =1 $ then $ a = b / \sigma(b) $ for some $b$ in $L$ and $ \sigma$ a generator of the Galois group.

My question: Suppose $N_{L / K}(a) \simeq 1 $, is there some $b$ such that $ a \simeq b /\sigma(b) $?

I guess I could have try to make the question more precise, but there seems to be some merits in leaving it a bit vague.

I have accept the the answer by Paul Broussous, which address the situation when the extension is unramified. It is because that is what I need. I am still curious whether something can be done when the extension is totally ramified?