Are norms of elements of cyclotomic fields positive?

Question

For any nonzero element in a cyclotomic field $\mathbb{Q}(\zeta_{n})$ with $n \geq 3$, is it true that its norm is positive? If so, how could I possibly show this?

This comes from Ireland and Rosen's chapter 13 question 3. I am currently working through this book - any hints or help is appreciated!

Answer 1

Let $S$ be a set of coset representatives for the subgroup of ${\rm Gal}({\mathbb{Q}(\zeta_n)/\mathbb{Q}})$ generated by complex conjugation. Then:

$$N_{\mathbb{Q}(\zeta_n)/\mathbb{Q}}(\alpha)=\prod_{g\in {\rm Gal}({\mathbb{Q}(\zeta_n)/\mathbb{Q}})} g\alpha=\left(\prod_{g\in S} g\alpha\right)\overline{ \left(\prod_{g\in S} g\alpha\right)} $$