1.6.88 Fields, Groups, Rings

Condition: let \ (\ mathbb {r} \) denotes an additive group of actual numbers, \ (\ varphi: \ mathbb {r} \ rightarrow \ mathbb (c}^{*} \) set by the formula \ (\ varphi (x) = e^^} {i x} \). Prove that \ (\ varphi- \) group homomorphism. Find his core and image. Is it \ (\ varphi \) monomorphism, epimorphism, isomorphism?