Question regarding function field

Question

I have learned in my algebraic curves class that the function field is the field of rational functions on a curve $C$ (or some variety).

I was at a number theory talk, where the person counted the number of solutions in $\mathbb{F}_q[t]$ of certain polynomial equations over $\mathbb{F}_q(t)$ and was calling it the "function field" setting.

I was wondering if they have the same name because there is some relation? or is it just a coincidence that they both call it the function field?

Thanks!

Answer 1

I don't know the actual etymology, but it is true that $\mathbf{F}_q(t)$ is the field of rational functions on the affine line $\mathbf{A}_{\mathbf{F}_q}^1$.