TITLE: An alternative description of K^*/Nm(L^*)
QUESTION [7 upvotes]: Is there a nice explicit description for the group $K^*/Nm_{L/K}(L^*)$ for a finite field extension $L/K$? 
What if for example, $L$ is obtained from $K$ by ajoining an n-th root of some $\alpha \in K$
(and assuming that $K$ contains the n-th root of unity)?
I don't see a nice answer even for the case $n=2$.
Thank you. 
EDIT: Thanks for the answers! Is that correct that in the case of a cyclic extension this group is isomorphic to $Br(L/K)$, since both these groups are identified with $H^2(Gal(L/K), L^*)$?

REPLY [4 votes]: In addition to Alex's answer I would like to point out that studying the group  $K^\times/N(L^\times)$ is a perfectly fine goal (related of course to the validity of the Hasse norm principle in extensions of number fields), which should not simply be dismissed as a "wrong question". A good place to start is the work by Leonid Stern, the most recent article being
On the norm groups of Galois $2\frak n$-extensions of algebraic number fields,
J. Number Theory 129, No. 5, 1191-1204 (2009).