Place (number field)

Summary

A place of an algebraic number field $K$ is a class of equivalent nontrivial absolute values on $K$. The finite places correspond to the nonzero prime ideals of the ring of integers. The infinite places correspond to the real embeddings and the pairs of complex conjugate embeddings of $K$. Each place gives a completion of $K$, which is a local field or $\R$ or $\C$. The places together are the index set of the adele ring. For $K = \Q$, Ostrowski's theorem says that the places are the primes $p$ and one infinite place.