Why we need polynomials to prove that $1,a,a^2,\dots , a^{n-1}$ is a field?

Question

I'm reading Lang's "Undergraduate Algebra", here:

It's not clear to me why proving that that is a field, we need to invoke polynomials. I'll also add the statement and proof of this previous theorem which I suspect may have something to do with it but I can't spot it.

Answer 1

here is the top of that page and previous