Neukirch claims in Lemma 10.1 of his ANT that $(1-\zeta)$ is a prime ideal of degree 1 in the ring of integers of $\mathbb{Q}(\zeta)$.
Here, I wasn't sure what prime ideal of degree 1 means - does he mean inertia degree (but then: over what other prime ideal?)
In my opinion, it should be the inertia degree over the prime ideal $l\mathbb{Z}$ in $\mathbb{Z}$, however, he doesn't state it in this way and only introduces this ideal later in the statement.
Here $\zeta$ is primitive $n$-th root of unity, where $n=l^{\nu}$ for some prime $l$.
As mentioned in the theorem, $l\mathcal{O}=(1-\zeta)^d$ where $d$ is the degree of the extension.
When you have a prime ideal in a ring of integers, you can get the corresponding prime ideal in $\mathbb{Z}$ by $l\mathbb{Z}=(1-\zeta)\cap\mathbb{Z}$. As you mentioned, the degree is indeed inertia degree over the obvious prime ideal $l\mathbb{Z}$ in $\mathbb{Z}$.