Does an infinite polynomial define algebraic numbers?

Question

As the title says, does a polynomial with an infinite number of terms define algebraic numbers as roots? An algebraic number is defined as a solution to a polynomial with rational coefficients, but it is not usually specified whether this polynomial can have infinite terms.

Answer 1

In view of the wording at the end of the question, it seems worthwhile to say explicitly that neither infinitely many terms nor infinite terms are possible in a polynomial.

With infinitely many terms, you'd get power series (not polynomials), and these can have roots that are not algebraic; for example, $\pi$ is a root of the sine function, which is given by an everywhere convergent power series.

As for infinite terms, I don't know what that would mean.

Answer 2

No, by definition a polynomial has finitely many terms, despite the fact that polynomials of every finite degree exist.

Anyway, the algebraic numbers are dense in $\mathbb{C}$ and so any continuous function that is zero at all algebraic numbers is actually just zero.