Definition
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[0XF] Each rational number \(x≠0\) breaks down uniquely as a product
\begin{equation} \label{primi} x=± p_ 1^{m_ 1}p_ 2^{m_ 2}\cdots p_ k^{m_ k}\ , \end{equation}
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where \(p_ 1{\lt}p_ 2{\lt}\cdots {\lt}p_ k\) are prime numbers and the \(m_ j\) integers. Fixed as above a prime number \(p\), we define the \(p\)–adic absolute value of \(x∈ℚ\) as
\[ |x|_ p=\begin{cases} 0 & \text{ ~ \text{if}~ ~ }x=0\\ p^{-m}& \text{ if $p^ m$ is the factor with base $p$ in the decomposition~ ~ \ref{primi}\ .} \end{cases} \]
Finally, we define \(d(x,y)=|x-y|_ p\), which will turn out to be a distance on \(ℚ\), called \(p\)–adic distance.