Rational Number - p-adic Numbers

p-adic Numbers

In addition to the absolute value metric mentioned above, there are other metrics which turn Q into a topological field:

Let p be a prime number and for any non-zero integer a, let |a|_p = p−n, where pn is the highest power of p dividing a.

In addition set |0|_p = 0. For any rational number a/b, we set |a/b|_p = |a|_p / |b|_p.

Then d_p(x,y) = |x − y|_p defines a metric on Q.

The metric space (Q,d_p) is not complete, and its completion is the p-adic number field Q_p. Ostrowski's theorem states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.