Let p be a number (generally, we take p to be a prime number, but a large part of the theory works without this condition). We may define a p-adic valuation vp: N -> N (considering to be a natural number, for simple convenience) as follows: vp(n) is the number of times p divides n. We may extend the definition to the rational numbers, vp: Q -> Z, by defining vp(n/m) = vp(n) - vp(m) (note that this is well defined!).

For convenience, we usually consider vp(0) = infinity, in the sense of real analysis.

These are all obvious properties of the p-adic valuation:

  • vp(a*b) = vp(a) + vp(b).
  • vp(pk) = k.
  • vp(a+b) >= min(vp(a), vp(b)).