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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, ...".
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} ).
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
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