I'm studying the construction of unramified extensions, and many references say that it's enough to attach the $p^n-1$ primitive root of unity to $\mathbb{Q}_p$ in order to obtain the unique degree $n$ unramified extension of the $p$-adics.
What I don't understand, is why the minimal polynomial of it, so a cyclotomic polynomial, has degree $n$. I'd say that the cyclotomic polynomial has degree $\phi(p^n-1)$, but $\phi(p^n-1)\neq n$ in general.
So where I'm overlooking something?
The document says to add a "primitive root of order $p^n-1$" not a "primitive $n$-th root of one". So it must be an irreducible factor of $x^{p^n-1}-1$ that has degree $n$, but not $x^n-1$.