How do I represent the elements of $\mathbb{F}_9$ over $\mathbb{F}_3$

Question

How do I represent the elements of $\mathbb{F}_9$ over $\mathbb{F}_3$?

The question hints to use the fact that $\mathbb{F}_9$ is the 8th cyclotomic field over $\mathbb{F}_3$, but I dont see how that helps. How does it help to know there are elements $a$ in $\mathbb{F}_9$ such that $a^8 = 1$ in $\mathbb{F}_3$?

Answer 1

$\mathbb{F}_9$ is the splitting field of every quadratic polynomial irreducible over $\mathbb{F}_3$.

The simplest such polynomial is $x^2+1$.

Let $\alpha \in \mathbb{F}_9$ be a root of $x^2+1$. Then $\mathbb{F}_9=\mathbb{F}_3(\alpha)=\mathbb{F}_3[\alpha]=\{ a+b\alpha: a,b \in \mathbb{F}_3 \}$, where $\alpha^2=-1$.

Alternatively, $(\alpha+1)^4=-1$ and so $\mathbb{F}_9^* = \{ (\alpha+1)^n : n=0,\dots,8 \}$