I am trying to understand how to show something is a primitive polynomial; I understand it has to be irreducible by definition, and according to Wolfram:
A primitive polynomial is a polynomial that generates all elements of an extension field from a base field.
For example, the polynomial $1 + x + x^2$ is primitive over $GF(2^3)$. What is the rudimentary way of generating the elements of $GF(2^3)$ from this polynomial? My knowledge is not in a math-focused field so I just need a straightforward methodology for generating the elements to show polynomials are primitive under $GF(2^m)$.