I am trying to understand the proof given in https://kconrad.math.uconn.edu/blurbs/gradnumthy/ostrowskinumbfield.pdf of the product formula:
In the proof, it says the following: "The formula is clear when $\alpha \in \mathcal{O}_K^{\times}$, since every term in the product is 1".
Is this correct? I understand why the terms corresponding to the non-archimedean valuations are 1, but I am not sure why the archimedean ones must be 1.
If $\tau : K \rightarrow \mathbb{C}$ is an embedding, why $|\tau(\alpha)|=1$?
Thanks in advance.