Can we apply Fundamental theorem of Algebra on entire, nonconstant functions?

Question

I have the following question:

Can we apply the Fundamental theorem of Algebra on entire, nonconstant functions $f:\mathbb{C}\to\mathbb{C}$?

We can write such $f$ as $f(z)=\sum_{k=0}^{\infty}a_kz^k$ with radius of convergence $R=\infty$. The Theorem holds for finite sums. I'm not sure, if $f(z)=e^z$ is a counterexample that we can't apply the Theorem generally.

Answer 1

As you noted, the fundamental theorem of algebra isn't directly applicable to entire functions, as shown by the example $f(z) = e^z$, but there is a substitute, namely

Picard's little theorem If $f$ is an entire, non-constant function, then the equation $f(z) = a$ has a solution for every $a \in \mathbb{C}$ with at most one exception.

The proof of this is fairly tricky though. See a good intermediate-level textbook in complex analysis.