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I am trying to prove the following assertion:

If an algebraic number field $K$ contains a complex root of unity, then the norm of every nonzero element of $K$ is positive.

I think this is supposed to be a simple application of the theory of geometry of numbers, but I can't see how to prove it.

Can someone please help me?

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    Thanks. I see it now. If$K$had a real embedding $\sigma$, then the group $W$ of roots of unity in K is isomorphic to the group $\sigma(W)$ of roots of unity in $\sigma(K)$. But the hypotheses imply |W| > 2, while $|\sigma(W)| = 2$ because $\sigma(W)$ is real.2011-02-04

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