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# 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

$|x|_{p}:=\begin{cases}(\frac{1}{p})^{n}\quad\mathrm{when}\,\,x\neq 0,\\ 0\quad\mathrm{when}\,\,x=0,\end{cases}$ |

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.

$\prod_{{p\,\mathrm{prime}}}|x|_{p}=|x|_{\mathrm{tr}}\quad\forall x\in\mathbb{Q}.$ |

## Mathematics Subject Classification

13A18*no label found*

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