On finite etale morphisms to affine non-singular curve

Question

Let $f:X \to Y$ be a finite etale morphism over $\mathbb{C}$, where $Y$ is a non-singular affine curve. Suppose further that $X$ is connected. Is $f$ going to be an isomorphism?

Answer 1

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The short answer is no. As an easy example, consider $X=Y=\Bbb{C}^*$ and $f(x)=x^2$. - Mohan