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? 



Do finite places of a number field also correspond to embeddings?
The Archimedean places of a number field K do not quite correspond to the embeddings of K into $\mathbb{C}$: there are exactly $d = [K:\mathbb{Q}]$ of the latter, whereas there are 
$r\_1 + r\_2$ Archimedean places, where:


if $K = \mathbb{Q}[t]/(P(t))$, then $r\_1$ is the number of real roots of $P$ and $r\_2$ is the number of complex-conjugate pairs of complex roots of $P$. In other words, $r\_1$ is the number of degree $1$ irreducible factors and $r\_2$ is the number of degree $2$ irreducible factors of $P(t) \in \mathbb{R}[t]$.


There is a perfect analogue of this description for the non-Archimedean places. Namely, the places of $K$ lying over the $p$-adic place on $\mathbb{Q}$ correspond to the irreducible factors of $\mathbb{Q}\_p[t]/(P(t))$; or equivalently, to the prime ideals in the finite-dimensional $\mathbb{Q}\_p$-algebra $K \otimes\_{\mathbb{Q}} \mathbb{Q}\_p$. 


More generally: if $L = K[t]/P(t)/K$ is a finite degree field extension and $v$ is a place of $K$ (possibly Archimedean), then the places of $L$ extending $v$ correspond to the prime ideals in $L \otimes\_K K\_v$ or, if you like, to the distinct irreducible factors of $P(t)$ in $K\_v$, where $K\_v$ is the completion of $K$ with respect to $v$.


See e.g. Section 9.9 of Jacobson's *Basic Algebra II*.


By coincidence, this is exactly the result I'm currently working towards in a course I'm teaching at UGA. I'll post my lecture notes when they are finished. (But I predict they will bear a strong resemblance to the treatment in Jacobson's book.)


