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

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)).