# N roots?

Homework Helper
The function $$y=e^x$$ can be expanded using the power series, thus $$y=e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3!}+...$$
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 $e^x=0$ there are zero roots. Why is this possible when, clearly by the power series for $e^x$ it should be an infinite degree polynomial with infinite roots (in the complex plane)?

Borek
Mentor
According to wiki:

In mathematics, a polynomial is an expression of finite length

so perhaps that's where the problem lies.

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

Borek
Mentor
No idea.

Actually, I have checked other source - Wolfram Mathworld - and their definition of polynomial doesn't state it has to be finite.

HallsofIvy
The phrase "polynomial of infinite degree" is never never used by real mathematicians. In particular $$e^z$$ is not (repeat not) a polynomial in $$z$$.