generating set for finite fields

Question

let us consider GF(2^n) as a vector space over GF(2), Is it possible to find a generating set for GF(2^n)? How can ew find it? I want to define a linear transformation of GF(2^n) to itsefl.

Answer 1

For variety, a different approach to defining $GF(2)$-linear endomorphisms of the field $GF(2^n)$ is to observe that the ring of all such endomorphisms is generated by $GF(2^n)$ (which acts by multiplication) and the Frobenius endomorphism $x \mapsto x^2$.