Number of forms 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 2⁸a'1 different non-zero vectors that you can map the first basis vector to. Then there are 2⁸a'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 2⁸a'2^ka'1 aren't, so the total number of different automorphisms is

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

• The sum-product theorem for finite fields
• Entropy extractor
• Inside the Rijndael (AES) S-box