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?
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Asaf Karagila
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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.
Arthur
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A module over a field is a vector space. Why call it "module" then? – enedil Mar 14 '19 at 08:57
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@enedil Because once we get to this stage, the word "vector" is already in heavy use. Having one more thing called a "vector space" and an entirely new set of "vectors" to keep track of as we're working will not help. Also, it's convention. And as you say, it is exactly the same thing in this case, so we don't lose anything. – Arthur Mar 14 '19 at 09:01