- #1

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## Homework Statement

##(\mathbb{C}^\times,\cdot)/\mu_m\cong (\mathbb{C}^\times,\cdot)## for any integer ##m\geq 1##, where ##\mu_m=\{z\in \mathbb{C} \mid z^m=1\}##.

## Homework Equations

## The Attempt at a Solution

Here is my idea. Consider the map ##f: \mathbb{C}^{\times} \to \mathbb{C}^{\times}## where ##f(z) = z^m##. Then certainly ##\ker(f) = \mu_m##. But since this kernel is nontrivial, that means that the map can't be surjective (since if f were surjective we would have to have a trivial kernel). This would mean that we couldn't establish the result using the first isomorphism theorem. Where am I going wrong?