Finite fields of size (order) a prime power (with exponent greater than 1) are constructed by first choosing an irreducible polynomial to reduce by. However, all finite fields of a given size are isomorphic: it doesn't matter which valid irreducible polynomial you choose to be the reduction polynomial.
Explain this isomorphism in detail. It's not just any permutation.
The zero (additive identity) and one (multiplicative identity) always stay the same. The "scalars", the result of repeatedly adding one to itself, stay the same. The "structure" (not really defining this) of "scalar" multiplication probably stays the same. There's probably more.
Algorithmic problem: given two finite fields of the same size specified by their addition and multiplication tables, determine the permutation that maps between them.
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