abstract algebra - Finite fields and primitive elements

Let $\mathbb F_9$ be a finite field of size $9$ obtained via the irreducible polynomial $x^2 + 1$ over the base field $\mathbb F_3$.

1. How can you find a primitive element?


2. Make a list of the elements of $\mathbb F_9$ together with a primitive element and all the powers of the primitive element.

Answer

1. I assume that you are looking for a generator of $F^*$. You can just go through all the $8$ elements of $F^*=\{1,-1,x,x+1,x-1,-x,-x+1,-x-1\}$ and compute their multiplicative orders. But with a little bit of thought you can avoid most of these computations. The following method is also applicable in other finite fields. We have the Frobenius automorphism $a \mapsto a^3$. Remark that $a$ is a generator iff the order is $8$ iff $a^4=-1$. This already excludes $1,-1,x,-x$. So we should try $a=x+1$, and compute $a^3=x^3+1=x(-1)+1$, and $a^4=(-x+1)(x+1)=1-x^2=-1$. This shows that $x+1$ is a generator. The other generators are $x-1,-x+1,-x-1$.




2. Again you can do this easily with the help of the Frobenius.