p-adic valuation

Let $p$ be a positive prime number. For every non-zero rational number $x$ there exists a unique integer $n$ such that $$x = p^n\cdot\frac{u}{v}$$ with some integers $u$ and $v$ indivisible by $p$ . We define

obtaining a non-trivial non-archimedean valuation, the so-called $p$ -adic valuation $$|\cdot|_p:\,\mathbb{Q} \to \mathbb{R}$$ of the field $\mathbb{Q}$ .

The value group of the $p$ -adic valuation consists of all integer-powers of the prime number $p$ . The valuation ring of the valuation is called the ring of the p-integral rational numbers; their denominators, when reduced to lowest terms, are not divisible by $p$ .

The field of rationals has the 2-adic, 3-adic, 5-adic, 7-adic and so on valuations (which may be called, according to Greek, dyadic, triadic, pentadic, heptadic and so on). They all are non-equivalent with each other.

If one replaces the base number $\frac{1}{p}$ by any positive constant $\varrho$ less than 1, one obtains an equivalent $p$ -adic valuation; among these the valuation with $\varrho = \frac{1}{p}$ is sometimes called the normed $p$ -adic valuation. Analogously we can say that the absolute value is the normed archimedean valuation of $\mathbb{Q}$ which corresponds the infinite prime $\infty$ of $\mathbb{Z}$ .

The product of all normed valuations of $\mathbb{Q}$ is the trivial valuation $|\cdot|_\mathrm{tr}$ , i.e. $$\prod_{p\,\mathrm{prime}}|x|_p = |x|_\mathrm{tr} \quad \forall x\in\mathbb{Q}.$$