Number of representations of a finite field

The number of elements in a finite field must be a prime power, and for every prime power there is only one finite field up to isomorphism.

The finite field with 256 elements, GF(2⁸), is important in applications. From one perspective, there is only one such field. But there are a lot of different isomorphic representations of that field, and some are more efficient to work with that others.

Just how many ways are there to represent GF(2⁸)? Someone with the handle joriki gave a clear answer on Stack Exchange:

There are 2⁸−1 different non-zero vectors that you can map the first basis vector to. Then there are 2⁸−2 different vectors that are linearly independent of that vector that you can map the second basis vector to, and so on. In step k, 2^k-1 vectors are linear combinations of the basis vectors you already have, so 2⁸−2^k−1 aren’t, so the total number of different automorphisms is

$\prod_{k=1}^8\left(2^8 - 2^{k-1} \right ) = 5348063769211699200$

This argument can be extended to count the number of automorphism of any finite field.

Number of forms of a finite field

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