I have a finite field $\mathbb{F}_{27}$. I need to find an element of order 13. I know that multiplicative group of this field is cyclic with order 26. So I want to find a generator $g$ of this cyclic group, then $g^2$ will have order 13. Also I know that all elements of this field can be represented by polynomials from $\mathbb{F}_3[t]$, where t is root of irreducible polynom $x^3-x-1$.
Probably we can find generator of $\mathbb{Z}_{26}$ and than build an isomorphism to $\mathbb{F}_{27}\setminus 0$.
Anyway I don't know were to start.
Thanks!
Best Answer
Start with finding the order of, say, $t$. It can't be $2$ (since $t^2-1\neq0$), so it's either $13$ or $26$. So either $t$ or $t^2$ is the one you're looking for.