Can nonzero polynomials vanish identically?

Solution 1:

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.

Solution 2:

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.