Does a pair of complex embeddings correspond to a single complex prime?

Question

I'm reading through Neukirch's Algebraic Number Theory, and he distinguishes between a number field's real primes, given by its real embeddings, and its complex primes, which are induced by the pairs of complex conjugate non-real embeddings. What I'm confused about is whether the pair of complex embeddings is a single complex prime, or does the pair of complex embeddings correspond to a pair of complex primes?

Answer 1

Neukirch defines a prime as a class of equivalent valuations [(1.1), page 184]. A pair of complex conjugate embeddings lead to equivalent valuations, and thus represent a single prime. This is clear, since if $\tau$ and $\overline{\tau}$ are the embeddings, then the corresponding valuations $|\tau(x)|$ and $|\overline{\tau}(x)|$ are identical.