Something that seems to be pretty standard in every introductory treatment is that the infinite places correspond to embeddings into $\mathbb{C}$. Do the finite places correspond to embeddings as well? I can envision two possibilities. My first guess is that the primes sitting above $p \in \mathbb{Q}$ correspond to embeddings into $\overline{\mathbb{Q}_p}$, and thus also to embeddings into $\mathbb{C}$ by some messy non-canonical field isomorphism. My second guess, which I think would imply the first, is that the places of $\mathbb{Q}[\alpha]$ above $p \in \mathbb{Q}$ correspond to embeddings into $\mathbb{Q}_p[\alpha]$. I've never been able to find a precise statement about this in any of the texts I've been studying (mostly Milne's notes and Frohlich & Taylor) and would appreciate if anyone could let me know where to learn more about this -- or if I'm just plain wrong.

One other thing is that the embeddings into $\mathbb{C}$ play a central role in analyzing the basic structure of a number field by way of Minkowski theory. Is there some analog for the finite places, or does that even make any sense?