Product Formula for Norms on Non-zero Rationals/Lemma

Theorem
Let $\Q$ be the set of rational numbers.

Let $\Bbb P$ denote the set of prime numbers. Let $z \in \Z_{\ne 0}$.

Then the following infinite product converges:
 * $\size z \times \ds \prod_{p \mathop \in \Bbb P}^{} \norm z_p = 1$

where:
 * $\size {\,\cdot\,}$ is the absolute value on $\Q$
 * $\norm {\,\cdot\,}_p$ is the $p$-adic norm on $\Q$ for prime number $p$

Case 1 : $z \in \Z_{>0}$
Let $z \in \Z_{>0}$.

From Fundamental Theorem of Arithmetic, we can factor $z$ as a product of one or more primes:
 * $z = p_1^{b_1} p_2^{b_2} \dots p_k^{b_k}$

Then for every prime number $q$:
 * $\norm z_q = \begin{cases}

p_i^{-b_i} & : \exists i \in \closedint 1 k :q = p_i \\ 1 & : \forall i \in \closedint 1 k : q \ne p_i \\ \end {cases}$

By definition of absolute value on $\Q$:
 * $\size z = p_1^{b_1} p_2^{b_2} \dots p_k^{b_k} $

For $n \ge \max \set{p_1, p_2 \dots p_k}$:

Hence:

Case 2 : $z \in \Z_{<0}$
Let $z \in \Z_{<0}$.

Hence
 * $-z \in \Z_{>0}$.

We have:

In either case:
 * $\size z \times \ds \prod_{p \mathop \in \Bbb P} \norm z_p = 1$