Embedding of a number field into a p-adic algebraic closure ($\iota :K\hookrightarrow {\overline{\mathbb{Q}}}_p$)

Such a field embedding pulls the p-adic absolute value back to a finite place of a number field above $p$. It extends to an embedding of the corresponding completion, whose image is a finite extension of the P-adic numbers.

Finite places of a number field above $p$ correspond to absolute Galois group orbits of embeddings of a number field into a p-adic algebraic closure. The absolute value is invariant under the group. Conversely, embeddings inducing the same place extend to embeddings of the same local completion and are conjugate.