Lemma 1   The cyclotomic units are algebraic units.

Proof. In order to prove the lemma, we will check that both $\eta$ and $\eta^{-1}$ are algebraic integers, thus $\eta$ is a unit. Notice that it suffices to prove that $\eta$ is an algebraic integer, because the rest follows from interchanging the role of $r$ and $s$ .

Let $r,s\in \Ints$ be relatively prime to $m$ , thus $r \mod m, s\mod m$ are units in $\Ints/m\Ints$ and we can find an integer $a$ such that: $$a\cdot s \equiv r \mod m$$ Note also that it follows that $\zeta^r=\zeta^{as}$ . Moreover, using the equality of polynomials: $$x^{as}-1=(x^s-1)\cdot(x^{s(a-1)}+x^{s(a-2)}+\ldots+x^s+1)$$ we get: \begin{eqnarray*} \eta &=& \frac{\zeta^r-1}{\zeta^s-1}=\frac{\zeta^{as}-1}{\zeta^s-1}\\ &=& \frac{(\zeta^s-1)\cdot(\zeta^{s(a-1)}+\zeta^{s(a-2)}+\ldots+\zeta^s+1)}{\zeta^s-1}\\ &=& \zeta^{s(a-1)}+\zeta^{s(a-2)}+\ldots+\zeta^s+1 \in \mathcal{O}_L=\Ints[\zeta] \end{eqnarray*}Hence the result.