product formula

《Algebraic Number Theory》 · 第 185 页 · Chapter III，Section 1「Primes」，Proposition 1.3

The normalized valuations $|\ |_{\mathfrak p}$ satisfy the following product formula, which demonstrates how important it is to include the infinite primes. (1.3) Proposition. Given any $a\in K^*$, one has $|a|_{\mathfrak p}=1$ for almost all $\mathfrak p$, and $\prod_{\mathfrak p}|a|_{\mathfrak p}=1$.