Root of irreducible polynomial not in general a generator

Question

Why is a root of an irreducible polynomial of degree 6 in $F_2[x]$ not in general a generator of $F_{64}^*$ ?

Answer 1

Let $K=GF_2$, $p\in K[x]$ be irreducible and of degree $6$.

If $a\in GF_{64}$ is any root of $p$, then the (distinct) roots of $p$ are $a,a^2,a^4,a^8,a^{16},a^{32}$. Then the subgroup of ${GF_{64}}^*$ generated by $a$ has $7,9,21$ or $63$ elements. Yet, $7$ is not convenient because, then $a^8=1$, and $a$ is in a subfield of $8$ elements (cf. the comment of ancientmathematician).

An example of each type:

If $p=x^6+x+1$, then $a$ is a generator of ${GF_{64}}^*$.

If $p=x^6+x^5+x^4+x^2+1$, then the subgroup has $21$ elements.

If $p=x^6+x^3+1$, then the subgroup has $9$ elements.