Proof of Dedekind's Theorem on the Galois Groups of rational polynomials Dedekind's theorem states that if a polynomial in $\mathbb Z[x]$ is factored into irreducibles modulo a prime not dividing the discriminant, then the Galois group of the polynomial, considered as a subgroup of $S_n$, contains a permutation whose cycle type corresponds with the degrees of the irreducible factors.
I've looked around a bit for a proof of this but couldn't find one. Can someone please point me to a proof?
 A: A quick proof depending on the basic theory of decomposition groups can be found on page 15 of these notes (see Corollary 2.7).
Edit: I decided to elaborate on what's included in the notes linked above.
As stated there, this result essentially follows from the surjectivity of the natural map
$$
    D_{\mathfrak{P}|\mathfrak{p}} = \{\sigma \in \text{Gal}(L/K) : \sigma(\mathfrak{P}) = \mathfrak{P}\} \to \text{Gal}((\mathcal{O}_L/\mathfrak{P})/(\mathcal{O}_K/\mathfrak{p}))
$$
for $L/K$ finite Galois extension of number fields, $\mathfrak{p}$ a prime of $K$, and $\mathfrak{P}$ a prime of $L$ above $\mathfrak{p}$.
To apply this here, one assumes $L$ is the splitting field over $K$ of a monic irreducible $f \in \mathcal{O}_K[X]$ and $\mathfrak{p}$ is a prime of $K$ modulo which $f(X)$ factors into a product of distinct irreducible polynomials. (So in particular, one can take $K = \mathbb{Z}$ and a prime $p$ of the desired form.) For such a prime $\mathfrak{p}$, choose any prime $\mathfrak{P}$ of $L$ over $\mathfrak{p}$ and let $S = \{\alpha \in L : f(\alpha) = 0\}$ (a set of $n = \deg(f)$ distinct elements of $L$). Then $D_{\mathfrak{P}|\mathfrak{p}}$ acts on $S$, and this action gives us a homomorphism $D_{\mathfrak{P}|\mathfrak{p}} \to H \leq S_n$. By the result quoted above, this $H \cong \text{Gal}((\mathcal{O}_L/\mathfrak{P})/(\mathcal{O}_K/\mathfrak{p}))$, and (as the Galois group of a finite extension of finite fields) the RHS is a cyclic group, say generated by some $\sigma$. More explicitly, if $f \pmod{\mathfrak p} = f_1 \cdots f_r$ for (by hypothesis) distinct monic $f_i$ over $\mathcal{O}_K/\mathfrak{p}$, then $\mathcal{O}_L/\mathfrak{P}$ is the splitting field of $f \pmod{\mathfrak{p}}$ over $\mathcal{O}_K/\mathfrak{p}$, it follows from generalities about Galois extensions that (under appropriate/obvious assumptions regarding the map into $S_n$) $\sigma$ corresponds to the permutation $(1,\ldots,d_1)(d_1+1,\ldots,d_2)\cdots(d_{r-1}+1,\ldots,d_r)$ where $d_i = \deg(f_i)$ for $1 \leq i \leq r$. Pullback $\sigma$ to an element $\text{Gal}(L/K)$ along the surjective homomorphisms $\text{Gal}(L/K) \to D_{\mathfrak{P}\mid\mathfrak{p}} \to H$ to reach the desired conclusion.
