What is the relationship between non-Archimedean places of infinite extensions of number fields and primes in the ring of integers?

Since $\mathcal O_L$ is the union of $\mathcal O_{L_i}$, where $L_i$ runs over all the finite subextensions, giving a prime ideal $\mathfrak p$ in $\mathcal O_L$ is the same as giving a compatible collection of primes ideals $\mathfrak p_i$ in each $\mathcal O_{L_i}$. (We set $\mathfrak p_i := \mathfrak p \cap \mathcal O_{L_i}$, and $\mathfrak p = \cup_i \mathfrak p_i$.)

Since $L$ is the union of the $L_i$, giving an absolute value $v$ on $L$ is the same as giving compatible absolute values $v_i$ on the various $L_i$. (Take $v_i$ to be the restriction to $L_i$ of $v$.)

Combining the preceding two remarks, we see that the bijection between primes ideals and non-Archimedean valuations in the case of finite extensions extends to a corresponding bijection in the case of infinite extensions.