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It really depends on what kinds of field $K$ and $L$ are. For example, if they are finite fields, then the quotient vanishes, since the norm is always surjective.

If $K$ is a local field with finite residue field and $L/K$ is abelian, then class field theory says that $K^\times/{\rm Norm}_{L/K}(L^\times)$ is isomorphic to the Galois group of $L/K$ (more generally, if $L/K$ is an arbitrary finite extension of local fields, then the quotient is isomorphic to the Galois group of the maximal abelian subextensions of $K$). This isomorphism is reasonably explicit and is described in any good exposition of class field theory. If $K$ is perfect, but not necessarily finite, or a local field with perfect residue field, then one can still say something reasonably explicit (see e.g. Serre's book "Corps Locaux").

If $K$ is a global field, then the situation is more complicated and less explicit, because the "right" object to look at, from the point of view of class field theory, is not the norm quotient you have written down, but the same with the multiplicative groups of the fields replaced by idèle class groups.

You need to tell us more about the fields in question to get a better answer.

Edit: to slightly generalise what I have said above, you can replace "finite field" by "quasi-finite field" throughout, and the statements will still hold.

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It really depends on what kinds of field $K$ and $L$ are. For example, if they are finite fields, then the quotient vanishes, since the norm is always surjective.

If $K$ is a local field with finite residue field and $L/K$ is abelian, then class field theory says that $K^\times/{\rm Norm}_{L/K}(L^\times)$ is isomorphic to the Galois group of $L/K$ (more generally, if $L/K$ is an arbitrary finite extension of local fields, then the quotient is isomorphic to the Galois group of the maximal abelian subextensions of $K$). This isomorphism is reasonably explicit and is described in any good exposition of class field theory. If $K$ is perfect, but not necessarily finite, or a local field with perfect residue field, then one can still say something reasonably explicit (see e.g. Serre's book "Corps Locaux").

If $K$ is a global field, then the situation is more complicated and less explicit, because the "right" object to look at, from the point of view of class field theory, is not the norm quotient you have written down, but the same with the multiplicative groups of the fields replaced by idèle class groups.

You need to tell us more about the fields in question to get a better answer.