Let $K$ be a number field with an Archimedean absolute value $|\cdot |$ and let $\bar{K}$ be the completion of $K$ wrt this valuation. Then $\bar{K}\cong \mathbb R $ or $\mathbb C$.
My question is:
Does this imply that the Archimedean places of $K$ correspond bijectively to the real embeddings $K\hookrightarrow \mathbb R$ and complex conjugate pairs of embeddings $K\hookrightarrow \mathbb C$?
Are there other ways to show this?
Thank you for your help!