Clearly finite groups are of immense value in physics and these are also substructures of fields. However I never came across any computations involving finite fields at university and so I never learned about them explicitly.
Are there some physical motivations to study finite fields/Galois fields?
Can I study these objects in a physical context?
I'm coming to ask this question because I'm interested in generating functions (in a physical contexts), and hence zeta-functions. I now and them come across things like the Weil conjectures which mathematicans seem to love, but in trying to understand these I see that I miss the background.