Requirement on Norm for units in Cyclotomic Fields

Question

Consider $\zeta_p$ a primitive $p^{th}$ root of unity. Prove that $\alpha \in \mathbb Z[\zeta_p]$ is a unit iff $N(\alpha)=\pm1$.

I'm not even sure where to start with this. I know that $N(1-\zeta_p^j)=p$ for $j=1,\ldots,p-1$ and $p=(1-\zeta_p)\cdots(1-\zeta_p^{p-1})$, with $[\mathbb Q(\zeta_p):\mathbb Q]=p-1$.

I also know that $\mathcal{O}_{\mathbb Q(\zeta_p)}=\mathbb Z[\zeta_p]$.

In addition, by definition of a unit, there exists $\alpha^{-1} \in \mathbb Z[\zeta_p]$ such that $\alpha \alpha^{-1}=1.$

I'm not quite sure how these go together to prove this.

Answer 1

Just use the multiplicative property for norm: $\alpha$ is a unit iff there exists $\beta$ such that $\alpha\beta=1$. Now $\mathrm N(\alpha\beta)=\mathrm N(\alpha)\mathrm N(\beta)=\mathrm N(1)=1$, hence $\mathrm N(\alpha)=\pm 1$.

Conversely, if $\mathrm N(\alpha)\neq\pm 1$, $\alpha$ can't be a unit.

Answer 2

The fact about $1-\zeta_p$ you stated does not seem to help, as it is obviously not a unit (that would contradict the statement we are trying to prove!).

Instead, you might want to use the multiplicativity of the norm, as well as the fact that $N(\alpha) \in \mathbb Z$ for $\alpha \in \mathbb Z[\zeta_p]$ (being a coefficient of the characteristic polynomial of $\alpha$). Similarly, all Galois conjugates of $\alpha$ will lie in $\mathbb Z[\zeta_p]$ again (being roots of the same polynomial).