I want to prove the following:
Let $k$ be a number field and $S$ a set of primes of $k$ containing the primes $S_p$ that lie over the rational prime $p$. Then the extension of $k$ by the group of $p$-th roots of unity, $k(\mu_p) | k$, is unramified outside $S$.
The first step is, that for $\mathfrak{p} \notin S$ we have $\nu_\mathfrak{p}(p) =0$. Therefore the extension of the localisations at $\mathfrak{p}$, $k_\mathfrak{p}(\mu_p) | k_\mathfrak{p}$, is unramified. Why? The next step is already the conclusion but I don't see it either.
It must be pretty easy, maybe someone can give me a hint? Is there an alternative way of proving the above statement?
Thank you, Tom
This is an elaboration on my comment. The minimal polynomial for $\zeta$ divides $x^p - 1$. This polynomial has distinct roots in any finite field of characteristic $\ell \neq p$ (proof: it is obviously coprime to its derivative since the only root of the derivative is $0$ for $\ell \neq p$), so at any prime away from $p$ your field extension is unramified.
You may find this document helpful to understand the general principle being used here, which is the key tool for computing the splitting behavior of primes in extensions of number fields:
http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/dedekindf.pdf