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A finite field is a field with a finite number of elements. For each prime power $q^k$, there is a unique (up to isomorphism) finite field with $q^k$ elements. Up to isomorphism, these are the only finite fields.
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Induction from split and non-split tori for GL_2 over a finite field
I don't have much time, but maybe the following can lead you to a solution. I'm sloppy too, writing $G$ both for the algebraic group (over some finite field $k$) and for the set of points $G(k)$.
Thi …