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I am looking at the proof of the product formula theorem:

For each $x \in \mathbb{Q}$, it holds $$\prod_{p \leq \infty} |x|_p=1$$

The proof starts by this:

It is enough to show it for each positive integer $x$.

Could you explain to me why when we show it for each positive integer $x$, then we have shown that it holds for each $x \in \mathbb{Q}$ ?

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Because the norm is multiplicative: $$\prod_p \left |\frac x y \right |_p = \left(\prod_p |x|_p\right) / \left(\prod_p |y|_p\right)$$

Just make sure that you understand why this formula is true for $x= -1$!

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