TITLE: infinite dimension representations of an abelian group
QUESTION [5 upvotes]: How do you show that an abelian group has no irreductible infinite dimension representation? Is it only true for a locally compact group? 
The only proof I found myself is to study maximal ideals of the group C*-algebra. Do you know a simple proof without operator algebras theory?

REPLY [3 votes]: Let $K=\mathbb{C}(t)$, the field of rational functions.
Then $K$ is an infinite dimensional irreducible complex representation of the multiplicative group $K^\times$.