There is a field of size 256, but it isn't Z/256Z.
"Even though all fields of size p are isomorphic to Z/pZ, for n ≥ 2 the ring Z/(p^n)Z (the ring of integers modulo p^n) is not a field. The element p (mod p^n) is nonzero and has no multiplicative inverse."
"Even though all fields of size p are isomorphic to Z/pZ, for n ≥ 2 the ring Z/(p^n)Z (the ring of integers modulo p^n) is not a field. The element p (mod p^n) is nonzero and has no multiplicative inverse."