# How to show that if K contains a complex root of unity, then every nonzero element of K has positive norm?

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.

Hint: the given hypotheses imply that $K$ has no real places, and hence the embeddings into $\mathbb{C}$ come in complex-conjugate pairs. – Pete L. Clark Feb 4 '11 at 3:19
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. – AWC Feb 4 '11 at 3:39