How to find all of generators in a finite fields How can I find all generators of a finite field?
For example in GF(2^3) and X^3 + x^2 + 1 as primitive polynomial.
I don`t want all of solutions. I need some hint and help to solve this problem.
Thanks a lot.
Ya Ali.

the multiplicative group of $GF_{2^3}$ has prime order, so that tells you how many generators it has. also note what is the only subfield of $GF_{2^3}$ (a generator cannot lie in a subfield). and finally note that there are precisely two irreducible polynomials of degree $3$ over $GF(2)$. try computing the first seven powers of $\alpha$ where $1+\alpha^2+\alpha^3=0$, expressing each as a linear combination of $1, \alpha, \alpha^2$ with coefficients in $GF(2)$
