[ColinWright]

Yes, but what he said was:

  : the set of numbers mod 256 form a field.

Without qualification it seems reasonable to assume the usual operations of addition and multiplication, and not to assume that the integers mod 256 are simply being used as an opaque labelling of the elements of a field such as you describe.

And the labelling is opaque, because while it's obvious what polynomial an integer gets mapped to, the multiplication depends on the particular g.