What would be the simplest way to prove that every element in $F_{128}$ is a primitive root except zero $(0)$ and the identity.
Well, clearly 0 can not be a primitive root, and i also know that $F_128$ has got 126 primitive roots since, $p - 1$, being fairly large when i choose $2^7$ immediately yields this result. now, do i have to use brute force by checking all the 126 elements and showing that they are primitive roots or is there a better way. If there is, can someone show it to me, because i could not figure it out - let alone find it. Thanks.
Note that $\Bbb F_{128}^\times$ is a group with 127 elements. Since 127 is prime, every one of the 126 non-trivial elements generates the whole group.