nt.number theory - 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.)