N roots?

The function [tex]y=e^x[/tex] can be expanded using the power series, thus [tex]y=e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+...[/tex]
This is a polynomial of infinite degree, and the theorem that says a polynomial must have at least one root in the complex field, and thus this extends to a polynomial of nth degree having n roots (not necessarily distinct).

However, for [itex]e^x=0[/itex] there are zero roots. Why is this possible when, clearly by the power series for [itex]e^x[/itex] it should be an infinite degree polynomial with infinite roots (in the complex plane)?

 

Was that definition placed before or after this exception was noticed?