Completion of a number field at a prime ideal ($K_{\mathfrak{p}}$)

For a number field $K$ and a nonzero prime ideal $\mathfrak{p}\subset {\mathcal{O}}_K$, the completion $K_{\mathfrak{p}}$ is the completion for the $\mathfrak{p}$-adic absolute value. It is a finite extension of ${\mathbb{Q}}_p$, where $p\mathbb{Z}=\mathfrak{p}\cap \mathbb{Z}$.

For a number field $K$ and a rational prime $p$, the natural diagonal map gives an isomorphism of ${\mathbb{Q}}_p$-algebras

If $L/K$ is a finite Galois extension of number fields, its Galois group acts transitively on the prime ideals of ${\mathcal{O}}_L$ lying above a fixed prime ideal of ${\mathcal{O}}_K$.

For a prime $\mathcal{P}$ of a finite Galois extension $L/K$ above $\mathfrak{p}$, the decomposition group is the stabilizer

For a finite Galois extension of number fields, choose a nonzero residue field generator at $\mathfrak{q}$ and lift it to an integral element $a$ whose residues at the other primes above $\mathfrak{p}$ are zero, using the Chinese remainder theorem. The orbit polynomial ${\prod }_{\sigma \in G}(T-\sigma a)$ has coefficients in the base ring of integers. Modulo $\mathfrak{q}$, its nonzero roots are precisely the orbit of $\overline{a}$ under the decomposition group. Its coefficients belong to $k_{\mathfrak{p}}$, so finite-field Frobenius automorphism permutes these roots. Consequently some decomposition-group element acts as Frobenius on the generator, and therefore on the entire residue field. Frobenius generates the finite-field Galois group, proving surjectivity.

Let $M/K$ be a finite Galois extension, $L=M^H$, and $D$ the decomposition group of a prime $\mathfrak{P}$ above $\mathfrak{p}$. The bijectionfollows from transitivity on primes for both $M/K$ and $M/L$.

If $M/K$ is the Galois closure of a finite extension $L/K$, a prime splits completely in $L$ exactly when it splits completely in $M$. Complete splitting means every right action of $D$ on $H\backslash G$ fixes each coset, so $D$ lies in the subgroup core of $H$. That core is trivial because the conjugates of $L$ generate $M$.

Restriction to completions gives a canonical isomorphismConsequently the number of primes above $\mathfrak{p}$ is the index of the decomposition group.

For a prime $\mathfrak{q}$ of a finite separable extension $L/K$ above $\mathfrak{p}$, completion preserves the ramification index and residue-field extension: