1.6.88 Fields, Groups, Rings

condition: Let \( \mathbb{R} \) denote the additive group of real numbers, \( \varphi: \mathbb{R} \rightarrow \mathbb{C}^{*} \) given by the formula \( \varphi(x)=e^{i x} \). Prove that \( \varphi- \) is a group homomorphism. Find its core and image. Is \( \varphi \) a monomorphism, an epimorphism, an isomorphism?