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