Quote by freefall111
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?

That looks like too much machinery (at least for my tastes); I would rather just argue that a polynomial of degree n has (n+1)st derivative = 0 identically, and ask whether that can happen for exp(z).
RGV