I have given the following elliptic curve over $\mathbb{F}_9 = \mathbb{F}_3[\zeta_8]$ (where $\min_{\mathbb{F}_3}(\zeta_8) = x^2 - x - 1$):
$$ E/\mathbb{F}_9: y^2 = \zeta_8 x^3 + x. $$ Now I would like to know how to bring this equation in Weierstrass form, i.e. in the form $y^2 = x^3 + Ax + B$ for some $A,B \in \mathbb{F}_9$. I thought about doing a substitution $y = \zeta_{16} y'$ for some element $\zeta_{16}$ with $\zeta_{16}^2 = \zeta_8$, so I could divide by $\zeta_8$ on both sides. However, I don't think that $\zeta_{16}$ is in $\mathbb{F}_9$.
Could you please help me here?
Let $\zeta = \zeta_8$. Multiplying both sides of $y^2 = \zeta x^3 + x$ by $\zeta^2$ yields $$ (\zeta y)^2 = \zeta^2 y^2 = \zeta^3 x^3 + \zeta^2 x = (\zeta x)^3 + \zeta (\zeta x) \, . $$ Letting $\tilde{x} = \zeta x$ and $\tilde{y} = \zeta y$, then $$ \tilde{y}^2 = \tilde{x}^3 + \zeta \tilde{x} \, . $$