I would like to understand the definition of a Tschrinhaus transformation.
Take the irreducible polynomial $p(X) = 2X^2 + 2 \in \mathbb{F}_3[X]$ and define $L := \mathbb{F}_3[X]/(p(X)) \cong \mathbb{F}_9$. So $L$ is a finite extension of $\mathbb{F_3}$.
Now Wikipedia states that $L = \mathbb{F}_3(\alpha)$ with $\alpha = X \text{ mod } p(X)$ and the task is to find different primitive elements $\beta$ such that $L = \mathbb{F}_3(\beta)$. Then the minimal polynomial of $\beta$ is called a Tschirnhaus transformation.
My question is: How to find the different primitive elements? Does this depends on the specific case? Can I just use all irreducible polynomials $q(X)$ of $\mathbb{F}_3[X]$ with $deg(q(X)) \leq deg(p(X))$?