Connection between roots modulo primes and decomposition groups

Question

I am re-learning some algebraic number theory. I want to understand the following:

Let $f\in\mathbb{Q}[x]$ have irreducible factorization $f(x)=g_1(x)\cdot...\cdot g_m(x)$ over $\mathbb{Q}$. Let $K$ be a splitting field of $f$ over $\mathbb{Q}$, and let $G=\operatorname{Gal}(K/\mathbb{Q})$. Finally, if $\alpha_i$ is a root of $g_i$, then let $A_i=\operatorname{Gal}(K/\mathbb{Q}(\alpha_i))$.

If $f$ has a root in $\mathbb{Q}_p$ for some rational prime $p$ (possibly ramified in $K$), then for some $i$, there exists a prime $\mathfrak{p}$ of $K$ lying above $p$ such that the decomposition group $D_{\mathfrak{p}}$ is contained in $A_i$.

I understand that a root in $\mathbb{Q}_p$ implies in particular that $f$ has a root mod $p$. I also know that, since $f$ has a root in $\mathbb{Q}_p$, one of the irreducible factors of $f$ does as well, say $g_i$. Thus we have some $\alpha_i\in\mathbb{Q}_p$ with $g_i(\alpha_i)=0$.

However, I don't see the connection between this and the decomposition group $D_{\mathfrak{p}}$.

Answer 1

Let $g$ be an irreducible factor of $f$ that has a root in $\mathbb{Q}_p$. Over $K$, $g(x) = (x - \alpha_1)\cdots(x - \alpha_n)$. For $i \in \{1, \dots, n\}$, take a prime $\mathfrak{p} \in \mathbb{Q}(\alpha_i)$ lying over $p$ and a prime $\mathfrak{P}$ in $K$ lying over $\mathfrak{p}$. We use the following facts.

1. The decomposition group $D_{\mathfrak{P}} \subset \text{Gal}(K/\mathbb{Q}(\alpha_i))$ is isomorphic to the Galois group of $K_{\mathfrak{P}}/\mathbb{Q}(\alpha_i)_{\mathfrak{p}}$.

2. $\mathbb{Q}(\alpha_i)_{\mathfrak{p}}$ is isomorphic to $\mathbb{Q}_p(\alpha')$ where $\alpha'$ is some root of $g$ (considered as a polynomial over $\mathbb{Q}_p$) (see Neukirch for a proof of this statement, or the second answer to this stack exchange question.)

3. If $g \in \mathbb{Q}[x]$ has a root $\alpha$ in $\mathbb{Q}_p$, then $\mathbb{Q}_p$ is isomorphic to the completion of $\mathbb{Q}(\alpha)$ at a prime $\mathfrak{p}$ lying above $p$. (This is a consequence of Krasner's lemma; the details are laid out nicely here.)

So, there exists $\alpha_i$ and $\mathfrak{p} \in \mathbb{Q}(\alpha_i)$ such that $\mathbb{Q}(\alpha_i)_{\mathfrak{p}} \cong \mathbb{Q}_p$. Taking a prime $\mathfrak{P} \subset K$ lying above $\mathfrak{p}$, we have $D_{\mathfrak{P}} \cong \text{Gal}(K_{\mathfrak{P}}/\mathbb{Q}(\alpha_i)_{\mathfrak{p}}) \cong \text{Gal}(K_{\mathfrak{P}}/\mathbb{Q}_p)$. Thus, $D_{\mathfrak{P}}$ is isomorphic to the decomposition group of $\mathfrak{P}$ considered not as a prime over $\mathfrak{p}$ but as a prime over $p$. This establishes the claim.