Re: p-adic numbers, valuation ring



On 31.10.2006 15:14, James wrote:
If Q is the field of rational numbers, and v_p is the p-adic valuation with respect to the positive prime integer p, then what is the valuation ring? That is, what is the ring {x in Q such that v_p (x) >= 0 }?

I don't think it's Z, (the inegers), but my book makes me believe it is. My book says : "In the case of the field of rational numbers Q and the p-adic valuation v_p with its completion Q_p, the numbers 0,1, ..., p-1 form a system of representatives R for the residue class field Z / pZ of the valuation, ...".

It is not Z, it is Z_<p>, the localization of Z at the prime ideal
<p>=pZ, i.e. all quotients r/s with r,s in Z coprime and s coprime to p.

The author's saying that the residue class field is Z / pZ throws me off. The residue class field is defined to be o / p, where o is the valuation ring and p is the maximal ideal of o. (i.e. o = {x in Q s.t. v_p (x) >= 0} and p = {x in Q s.t. v_p (x) > 0} ).

The prime ideal P (I think it is better to denote by capital P such that
we do not get confused with the prime number p we have started with) is
generated by p (considered as element in o).

But if x is in Q and v_p (x) >= 0, all that means is that when you take out all powers of p from the denominator and numerator of x, you are left with a positive power of p. What's left over can certainly be a rational number, so what am I getting confused?

Thank you for your help,

James
.