In the following examples it is best not to use the polynomial representation, as the meaning of changes between the examples. The monic irreducible polynomial over GF(2) is not primitive. Let be a root of this polynomial (in the polynomial representation this would be ), that is, . Now , so is not a primitive element of GF(28) and generates a multiplicative subgroup of order 51. Consider the field element (in the polynomial representation this would be + 1). Now . As all the roots of this primitive polynomial are primitive elements, is a primitive element of GF(28) ( and no smaller power does). GF(28) has 128 generators (see Number of primitive elements). Having as a generator for a finite field is beneficial for many computational mathematical operations.