# Rational numbers with denoms not divisible by a prime p mod I is isomorphic to Z_p

1. Dec 3, 2011

1. The problem statement, all variables and given/known data
Show that the ring of rational numbers whose reduced form denominator is not divisble by a prime, p, mod an ideal the set of elements of the above set whose numerators are divisible by p is isomorphic to Z_p

2. Relevant equations

3. The attempt at a solution
It seems very trivial: Use 1st homomorphism theorem with phi(a/b) = a(modp), but I am having a hard time showing that such a mapping is actually a homomorphism additively. I.E., phi(a/b + c/d) = phi(ad+bc/bd) = ad+bc mod(p) =/= a modp + b modp = phi(a/b) + phi (c/d).

I am stuck here and any help would be appreciated.

Thanks,

2. Dec 3, 2011

### micromass

Staff Emeritus
Re: Rational numbers with denoms not divisible by a prime p mod I is isomorphic to Z_

Instead of constructing an isomorphism directly, maybe you can show that it's a field with p elements. This would imply it as well.

3. Dec 4, 2011