It is well known that if $f(x)$ is a polynomial over $\mathbb Z$ then for every prime $p$ (not dividing the discriminant of $f$ (thanks to KConrad)) the Galois group of that polynomial mod $p$ over $\mathbb{F}\_p$ embeds into the Galois group of $f$ over $\mathbb{Q}$. Where can I find a (easy) proof of this fact?



Galois group of a polynomial modulo $p$
1. This result of Dedekind is *not* true for every prime $p$, but only for primes not dividing the discriminant of $f(x)$.
2. There is no “easy” proof for someone who knows only Galois theory (the setting where the result is usually first met). You can find a proof in Jacobson’s Basic Algebra I, attributed to Tate, that aims to be self-contained at the level of just Galois theory, but when I tried to read that while first learning Galois theory I could not understand what was going on. I mention Dedekind’s theorem as Theorem 4.13 [here](https://kconrad.math.uconn.edu/blurbs/galoistheory/galoisaspermgp.pdf) and after its statement I indicate where “elementary” proofs can be found in Jacobson and in Cox’s book on Galois theory. I eventually learned that the right context for understanding what this theorem is all about is within algebraic number theory: the real meaning of the theorem is surjectivity of the reduction map from the decomposition group at an unramified prime to the Galois group of the residue field extension at that prime. Without algebraic number theory that context makes no sense and I believe a proof outside that context will not be worthwhile. Do you know any algebraic number theory yet?
3. In algebraic number theory books (Samuel, Marcus, Lang, Janusz, Koch) the proof of surjectivity of the map from the decomposition group at a prime to the Galois group (or just automorphism group) of the residue field extension at that prime is overly complicated, relying on decomposition fields or completions. The proof of this result by Frobenius that I found by reading his Collected Works is much simpler and still short, so I wrote up that proof for myself [here](https://kconrad.math.uconn.edu/blurbs/gradnumthy/frobeniuspf.pdf) since it has been forgotten.


