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Since it is particularly easy to write down a basis of a Kummer extension $K=k(\mu)/k$ (where $\mu^n=a \in k$) as a $k$-vector space, I suspect that it is should not be terribly hard to write an explicit formula for a norm of an element in terms of its coordinates in this basis.

Is there a standard text where this derivation is carried out?

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    Ой, ясно, что такой результат не в индексе. Он не важен по сравнению с другой информацией в книге о норме (см. алг. дополнение).2012-05-09

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