# independence of $p$-adic valuations

We prove the following particular case:

###### Proposition 1.

Let ${p}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{p}_{n}\mathrm{\in}\mathrm{Z}$ be distinct prime numbers^{} and let $\mathrm{\mid}\mathrm{\cdot}{\mathrm{\mid}}_{{p}_{i}}$ be the corresponding $p$-adic valuations^{} of $\mathrm{Q}$. Let ${a}_{\mathrm{1}}\mathrm{,}\mathrm{\dots}\mathrm{,}{a}_{n}\mathrm{\in}\mathrm{Z}$ and let ${\u03f5}_{i}$ be arbitrary positive real numbers, then there exists $y\mathrm{\in}\mathrm{Z}$ such that for all $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots}\mathrm{,}n$:

$$ |

###### Proof.

Let $p$ be an arbitrary prime, and let $\u03f5$ be an arbitrary positive real number. Notice that $\mathbb{Z}$ injects into ${\mathbb{Z}}_{p}=\underleftarrow{\mathrm{lim}}\mathbb{Z}/{p}^{n}\mathbb{Z}$, the $p$-adic integers. For any $b\in \mathbb{Z}$, we also write $b$ for its image in ${\mathbb{Z}}_{p}$, and it can be written as a sequence^{} $b=({b}_{j})$ with $b\equiv {b}_{j}mod{p}^{j}$. Let $n={n}_{p,\u03f5}\in \mathbb{N}$ be such that $$ (and thus for any other $c\in \mathbb{Z}$ such that $c\equiv {b}_{n}mod{p}^{n}$ we have $$).

Now, for the proof of the proposition^{}, let ${n}_{i}={n}_{{p}_{i},{\u03f5}_{i}}$ and recall that by the Chinese Remainder Theorem^{} we have an isomorphism^{}:

$$\prod _{i=1}^{n}\mathbb{Z}/{p}_{i}^{{n}_{i}}\mathbb{Z}\equiv \mathbb{Z}/(\prod {p}_{i}^{{n}_{i}})\mathbb{Z}$$ |

Therefore we can find an element $\stackrel{~}{y}$ of $\mathbb{Z}/(\prod {p}_{i}^{{n}_{i}})\mathbb{Z}$ (and thus a lift $y$ of $\stackrel{~}{y}$ to $\mathbb{Z}$) such that $y\equiv {a}_{i}mod{p}_{i}^{{n}_{i}}$ for all $i=1,\mathrm{\dots},n$. Hence:

$$ |

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Title | independence of $p$-adic valuations |
---|---|

Canonical name | IndependenceOfPadicValuations |

Date of creation | 2013-03-22 14:12:14 |

Last modified on | 2013-03-22 14:12:14 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 4 |

Author | alozano (2414) |

Entry type | Corollary |

Classification | msc 11R99 |

Related topic | Valuation |

Related topic | PAdicIntegers |

Related topic | PAdicValuation |