There is a field with 256 elements, because 256 is the power of a prime. But that field is not the integers mod 256: it has different rules for addition and multiplication.
There are also finite (Galois) fields GF(p^n) of size p^n (positive integer powers of p)
These can be exhibited by polynomials with coefficients in the GF(p) field with up to n terms.
Eg for p = 2 and n = 3
0
1
x
1 + x
1 + x + x^2
1 + x^2
x + x^2
x^2