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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.
5
votes
Representation of GL(n, F_p) over F_p, for n small
(For simplicity I will discuss $G = SL_n$. Making the necessary modifications to $GL_n$ shouldn't be hard.)
The most important result here is Steinberg's restriction theorem, which tells you that the …