Product Formula

Let $\alpha$ be a Nonzero Rational Number $\alpha=\pm {p_1}^{\alpha_1}{p_2}^{\alpha_2}\cdots{p_L}^{\alpha_L}$, where $p_1$, ..., $p_L$ are distinct Primes, $\alpha_l\in\Bbb{Z}$ and $\alpha_l\not=0$. Then

$$\vert a\vert \prod_{p{\rm\ prime}} \vert\alpha\vert _p = {p_... ...} {p_1}^{-\alpha_1}{p_2}^{-\alpha_2}\cdots{p_L}^{-\alpha_L}=1.$$

References

Burger, E. B. and Struppeck, T. ``Does $\sum_{n=0}^\infty {1\over n!}$ Really Converge? Infinite Series and $p$-adic Analysis.'' Amer. Math. Monthly 103, 565-577, 1996.