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On what Galois field AES really works?

I’m trying to understand the GF theory, but every time I come across information about AES it all makes no sense.

In my opinion $GF(2^8)$ defines any polynomial of the form:

$a_{7} x^7 + a_{6} x^6 + a_{5} x^5 + a_{4} x^4 + a_{3} x^3 + a_{2} x^2 + a_{1} x^1 + a_{0}$

Where $a_{i}$ can be 0 or 1. And everywhere I come across information that AES works just in $GF(2^8)$. But in AES $a_{i}$ are bytes, right? So $a_{i}$ can be every number from 0 to 7. And this mean that we have here $GF(8^8)$. And it has nothing to do with $GF(2^8)$.

Either I still don’t understand the GF or they’re all making some kind of simplification that’s so far from the truth, that it shouldn’t be done. So what Galois field AES really uses?

On what Galois field AES really works?

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