Definition:P-adic Norm

Definition
The p-adic metric is a metric on the set of real numbers which yields a different topology from the regular Euclidean Metric.

Let $p \in \left\{{2,3,5,7,11,13,17,\ldots}\right\}$ be any prime number.

For any nonzero integer $a \ $, define $ord_p(a) \ $ to be the highest power of $p \ $ which divides $a \ $.

For any $x \in \Q \ $ (the set of rationals) with numerator $a \ $ and denominator $b \ $, define $ord_p(x)=ord_p(a)-ord_p(b) \ $.

Define a map $|*|_p:\Q \to \R_+$ as


 * $|x|_p = \begin{cases} \tfrac{1}{p^{ord_p(x)}} & \mbox{if }x \neq 0, \\ 0 &\mbox{if }x = 0\end{cases}$

For any real number $x$ which is the limit of the Cauchy sequence $\left\{{x_1,x_2,...}\right\}$, define $|x|_p = \lim_{n \to \infty} |x_n|_p$.

$|*|_p$ forms a norm on the real numbers, which induces a metric by


 * $d(x,y) = |x-y|_p$

The real numbers under this metric is called $\Q_p$.