# independence of $p$-adic valuations

We prove the following particular case:

###### Proposition 1.

Let $p_{1},\ldots,p_{n}\in\mathbb{Z}$ be distinct prime numbers and let $\mid\cdot\mid_{p_{i}}$ be the corresponding $p$-adic valuations of $\mathbb{Q}$. Let $a_{1},\ldots,a_{n}\in\mathbb{Z}$ and let $\epsilon_{i}$ be arbitrary positive real numbers, then there exists $y\in\mathbb{Z}$ such that for all $i=1,\ldots,n$:

 $\mid y-a_{i}\mid_{p_{i}}<\epsilon_{i}$
###### Proof.

Let $p$ be an arbitrary prime, and let $\epsilon$ be an arbitrary positive real number. Notice that $\mathbb{Z}$ injects into $\mathbb{Z}_{p}=\varprojlim\mathbb{Z}/p^{n}\mathbb{Z}$, 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 $n=n_{p,\epsilon}\in\mathbb{N}$ be such that $p^{-n}<\epsilon$ (and thus for any other $c\in\mathbb{Z}$ such that $c\equiv b_{n}\mod p^{n}$ we have $\mid b-c\mid_{p}\leq p^{-n}<\epsilon$).

Now, for the proof of the proposition, let $n_{i}=n_{p_{i},\epsilon_{i}}$ 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 $\tilde{y}$ of $\mathbb{Z}/(\prod p_{i}^{n_{i}})\mathbb{Z}$ (and thus a lift $y$ of $\tilde{y}$ to $\mathbb{Z}$) such that $y\equiv a_{i}\mod p_{i}^{n_{i}}$ for all $i=1,\ldots,n$. Hence:

 $\mid y-a_{i}\mid_{p_{i}}<\epsilon_{i}$

Title independence of $p$-adic valuations IndependenceOfPadicValuations 2013-03-22 14:12:14 2013-03-22 14:12:14 alozano (2414) alozano (2414) 4 alozano (2414) Corollary msc 11R99 Valuation PAdicIntegers PAdicValuation