Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)We explore the relation between self extensions of simple representations of quantum affine algebras and the property of a simple representation being prime. We show that every nontrivial simple representation has a nontrivial self extension. Conversely, we prove that if a simple representation has a unique nontrivial self extension up to isomorphism, then its Drinfeld polynomial is a power of the Drinfeld polynomial of a prime representation. It turns out that, in the -case, a simple module is prime if and only if it has a unique nontrivial self extension up to isomorphism. It is tempting to conjecture that this is true in general and we present a large class of prime representations satisfying this homological property.27441671613645Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)EPSRC [EP/H000054/1][DMS-0901253]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)CNPq [306678/2008-0]EPSRC [EP/H000054/1][DMS-0901253]