Is there a non-trivial character on any locally compact Abelian group?

Question

Let $G$ be a locally compact Abelian group. Is there a non-trivial character on $G$?

indeed, i want the proof of the existence of a non-trivial character on a locally compact Abelian group.

Answer 1

For a LCA group, $Hom(G,\Bbb R/\Bbb Z)\neq 0$. For example, let $G$ be non trivial compact group. consider the following exact sequence $$Hom(G,\Bbb R/\Bbb Z)\to Ext(G,\Bbb Z)\to Ext(G,\Bbb R)=0$$ If $Hom(G,\Bbb R/\Bbb Z)=0 0$ then $Ext(G,\Bbb Z)\cong Ext(\Bbb R/\Bbb Z,\hat{G})\cong \hat{G}=0$ which is contradiction.