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
to be a natural number
, for simple convenience) as follows: vp
) is the number of times p divide
. We may extend
the definition to the rational number
, by defining vp
) = vp
) - vp
) (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)).