Let F be a finite field of order q and let α be a primitive element in F. Show that $α^r$ is a primitive element in F if and only if r is coprime to q−1.
Show that 2 is a primitive element in F11. Make a list of all the primitive elements in F11.
I am completely lost with these. Can someone kindly point me in the right direction? I know how to construct the finite field set where gcd(a,n) = 1 for any element in that set.
$F-\{0\}$ is a cyclic group of order $q-1$ generated by $\alpha$ so $\alpha^r$ is a generator if and only if $gcd(r,q-1)=1$. We deduce that if $gcd(r,q-1)=1$, $\alpha^r$ is a primitive element.
If $gcd(r,q-1)=d>1$, let $c={{n-1}\over d}$, $(\alpha^r)^c=1$ implies that $P(\alpha^r)=0$ where $P=X^c-1$, this implies that $\alpha^r$ is the splitting field of $X^c-1$.
For the other question, show that $2$ generates $F_11$ (compute the multiple of $2$ mod $11$) and determine the integers $r$ inferior to $10$ such that $gcd(r,10)=1$.