# 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

$${|x|}_{p}:=\{\begin{array}{cc}{(\frac{1}{p})}^{n}\hspace{1em}\mathrm{when}x\ne 0,\hfill & \\ 0\hspace{1em}\mathrm{when}x=0,\hfill & \end{array}$$ |

obtaining a non-trivial (http://planetmath.org/TrivialValuation) 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 (http://planetmath.org/Fraction) 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 (http://planetmath.org/EquivalentValuations) with each other.

If one replaces the number $\frac{1}{p}$ by any positive $\varrho $ less than 1, one obtains an equivalent^{} (http://planetmath.org/EquivalentValuations) $p$-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 $\mathrm{\infty}$ of $\mathbb{Z}$.

The product^{} of all normed valuations of $\mathbb{Q}$ is the trivial valuation $|\cdot {|}_{\mathrm{tr}}$, i.e.

$$\prod _{p\mathrm{prime}}{|x|}_{p}={|x|}_{\mathrm{tr}}\mathit{\hspace{1em}}\forall x\in \mathbb{Q}.$$ |

Title | p-adic valuation |
---|---|

Canonical name | PadicValuation |

Date of creation | 2013-03-22 14:55:50 |

Last modified on | 2013-03-22 14:55:50 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 14 |

Author | pahio (2872) |

Entry type | Definition |

Classification | msc 13A18 |

Synonym | $p$-adic valuation |

Related topic | IndependenceOfPAdicValuations |

Related topic | IntegralElement |

Related topic | OrderValuation |

Related topic | StrictDivisibility |

Defines | $p$-integral rational number |

Defines | normed $p$-adic valuation |

Defines | normed archimedean valuation |

Defines | dyadic valuation |

Defines | triadic valuation |

Defines | pentadic valuation |

Defines | heptadic valuation |