1. The problem statement, all variables and given/known data

Prove that the analytic function e^z is not a polynomial (of finite degree) in the complex variable z.


3. The attempt at a solution

The gist of what I have so far is suppose it was a finite polynomial then by the fundamental theorem of algebra it must have at least one or more roots. e^z can never equal zero for hence this is a contradiction.

Is it okay for me to apply the fundamental theorem of algebra like this or am I kind of using a bit too much machinery here?