There are non trivial homomorphism from $\Bbb R/\Bbb Z$ into $\Bbb Z$

Question

I think... For an irrational number $r$, the map $f(r+\Bbb Z)=1$, and $f(a+\Bbb Z)=0$, for a not in the subgroup generated by $r+\Bbb Z$ generates ahomomorphism from $\Bbb R/\Bbb Z$ into $\Bbb Z$. Am I Right?

Answer 1

There is no such nontrivial homomorphism. Otherwise we would get a nontrivial homomorphism $f:\mathbb{R}\to\mathbb{Z}$ (by composing with the quotient map $\pi:\mathbb{R}\to\mathbb{R}/\mathbb{Z}$).

And such map cannot exist because if $f(x)=a\neq 0$ then what would $f\big(x/(2\cdot a)\big)$ be? You can apply the same argument to your map to see where it fails.