TITLE: Can nonzero polynomials vanish identically?
QUESTION [14 upvotes]: I know that a nonzero single-variable polynomial over a finite field can vanish identically e.g. take the product $\prod_a(x-a)$ for every $a$ in the field. But I know that for an infinite field this cannot happen since a degree $d$ polynomial has at most $d$ roots. My questions are:

Why does a nonzero two-variable or higher polynomial over $\mathbb{R}$ not vanish identically? (In this case I know they can't but I don't know why)
What about nonzero multivariate polynomials over other infinite fields?

REPLY [4 votes]: For the first part, view a two-variable polynomial in x and y as a single variable polynomial in x, with y as a parameter. So if we treat y as fixed, and let x vary, then it is identically zero if and only if all the coefficients (which are polynomials in y) are zero. But seeing as the coefficients are themselves single variable polynomials in y, they are identically zero if and only if all their coefficients are zero, i.e. we have the zero polynomial.