Is vector field a field (in the algebraic sense) also?

Question

This is something about basic confusion.

Let $M$ be a smooth manifold of dimension $n$.

We denote $X$ to mean the vector field.

Is the vector field $X$, a field (in the sense of abstract algebra) also?

Answer 1

A single vector field is simply a choice of vector for each point in the plane. It doesn't have "elements" in any sensible way, and especially not elements one can add or multiply.

The collection $\mathscr X$ of all vector fields on $M$ (or all continuous vector fields, or all smooth vector fields) certainly has algebraic structure, as they can be added, and they can be multiplied by (real or complex) scalars. We call such a structure a module (in this case it isn't wrong to call $\mathscr X$ a vector space either). But vector fields cannot in any sensible way be multiplied with one another, so such a set cannot be a ring, and therefore cannot be a field.