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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.

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Finite fields are essential for constructing MUBs (see this paper by Wootters and Fields). Also, dealing with quantum error correcting codes (for qubits) is essentially the same as doing linear algebra over GF(4) because of the way Pauli matrices behave (see arXiv:0904.2557 or arXiv:quant-ph/9608006).

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I cannot answer for the zeta-functions or Weil-conjectures but one Finite field comes into sight as answer to your question.

Quaternions discovered by Hamilton have four unity vectors with a strange multiplication. Frobenius did work on this four element pseudo field.

It is a almost field but lacks commutativity A x B = B x A

There is one real unity vector and three imaginary ones.

The three a part build vectors (not discussing the imaginary nature of the unity vectors) and are used in threedimensional geometry and quantum mechanics.

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Thanks for bumping the thread, but quaternions are no finite field. They contain a subset isomorphic to the reals and these are not even countable. The four is the dimension of the quaternions as a vector space. If you think about it, by your definition, the real numbers are a field with one element. PS: I did link to the finite fields Wikipedia page in the question. – NikolajK Sep 27 '13 at 22:01
@NickKidman This answer applies to the quaternion group, which is indeed finite but is of course not a field. – Emilio Pisanty Sep 27 '13 at 22:28
@EmilioPisanty: There is a nice list of the small finite groups Wikipedia article. There is also a mad person who makes a personal wiki on groups and their properties. – NikolajK Sep 27 '13 at 23:05

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