# When is a gradient vector field also an algebraic field?

I was thinking about the inconsistencies in mathematical vocabulary today, and I came to this simple, open-ended question: When is a vector field (or a gradient field) an honest algebraic field? Explicitly, the gradient version is stated:

Given some functionn $F: \mathbb{R}^n \rightarrow \mathbb{R}$, when does the set $\{\nabla F(\bar{x}) \ | \ \bar{x} \in \mathbb{R}^n\}$ form a field on $\mathbb{R}^n$? Specifically, what are the conditions on $F$, and under what type of vector "multiplication" does this form a field? (here I assume we would use component-wise addition)

Thanks!