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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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The standard procedure would be to write the norm as a determinant. There are quite a few books that introduce the norm in this way. –  franz lemmermeyer May 7 '12 at 19:32
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I don't agree that just because it is easy to write down a basis that it is supposed to be easy to write down an explicit formula for the norm in that basis. Have you tried already to write down a general norm formula for an extension of degree 3 or 4 generated by a cube root or fourth root of a number in the base field $k$? (Whether or not the cube or fourth roots of unity are in $k$ is irrelevant. Just assume that $K = k(\sqrt[n]{a})$.) It is quite a mess in degree 4. The degree 3 case has an explicit answer at the end of Section 1.2 of Borevich and Shafarevich's Number Theory. –  KCd May 7 '12 at 20:28
    
Kcd, I couldn't find the passage from Borevich-Shafarevich that you refer to. Franz Lemmermeyer, you are right, writing down the matrix of multiplication by an element of $K$ is straightforward. The expression of norm as the determinant of it is probably as explicit as one can get. –  Dima Sustretov May 8 '12 at 13:17
    
@Dima, Sorry, it is Section 1.3, not 1.2, and I should've said it is in Chapter 2 (other chapters have their own section 1.1, 1.2,...). See page 83 of the English version and page 99 of the Russian version. –  KCd May 8 '12 at 22:33
    
Kcd, thank you! I tried to find the passage looking through index, but it looks like it is incomplete. –  Dima Sustretov May 9 '12 at 1:37
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