# Can nonzero polynomials vanish identically?

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:

1. 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)
2. What about nonzero multivariate polynomials over other infinite fields?
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Have you tried induction on the number of variables? If your polynomial has $n$ variables, view it as polynomial with coefficients from the polynomial ring in $n-1$ variables. By induction hypothesis at least one of the coefficients does not vanish for some choice of values. Then use the base case, $n=1$, that you already know to justify the induction step. – Jyrki Lahtonen Feb 25 '13 at 12:39

Let $F$ be an infinite field, and $f(x, y) \in F[x, y]$ a nonzero polynomial.
Regard $f$ as a polynomial $g(y) = f(x, y) \in (F(x))[y]$. This is a polynomial in $y$, with coefficients in the infinite field $F(x)$. Since it has a finite number of distinct roots, there is a $b \in F$ such that $0 \ne g(b) = f(x, b) \in F[x]$. Now apply the result for the univariate case.