 Solution

In the absolute-value description, a place of a number field is an equivalence class of nontrivial absolute values, where equivalent absolute values give the same topology, or differ by a positive power. More generally this defines the rank-one absolute-value places used in weak approximation for a field. For a number field, finite places correspond to nonzero prime ideals of its ring of integers, while infinite places correspond to real embeddings and pairs of conjugate complex embeddings. The completion at $v$ is denoted $K_v$.

A valuation place can also be described by a valuation ring ${\mathcal{O}}_v\subset K$ and its residue map into $k_v$, extended by sending elements outside that ring to infinity. The finite places above are the discrete-valuation instances of this description.

The weak approximation theorem states that for finitely many pairwise inequivalent nontrivial absolute values $v_1,\dots ,v_r$, the diagonal image of $K$ is dense in ${\prod }_i{K}_{v_i}$. Equivalently, for arbitrary $a_i\in K_{v_i}$ and ${ϵ}_i>0$, there is $a\in K$ such thatThe places must be distinct: two requirements at the same completion need not be compatible. In the number-field setting this permits simultaneous prescribed finite congruences and archimedean approximations.