p-adic valuation (idea)

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 v_p: N -> N (considering 0 to be a natural number, for simple convenience) as follows: v_p(n) is the number of times p divides n. We may extend the definition to the rational numbers, v_p: Q -> Z, by defining v_p(n/m) = v_p(n) - v_p(m) (note that this is well defined!).

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

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

• v_p(a*b) = v_p(a) + v_p(b).
• v_p(p^k) = k.
• v_p(a+b) >= min(v_p(a), v_p(b)).