equation over $\mathbb{Z}_3$

Question

Consider the ring $\mathbb{Z}_3$ of 3-adic integers. Does there exist a positive integer $n$ and a solution to $(X_1^2 + X_2^2 + \cdots + X_{n - 1}^2)^2 = 2X_n^4$ in $\mathbb{Z}_3^n$? If so, what is the smallest $n$ for which a solution exists?

Answer 1

There are no solutions with $X_n \neq 0$, for then dividing through by $X_n^4$ would yield that $2$ is a square in $\mathbb{Q}_3$...which it isn't.

Solutions with $X_n = 0$ correspond to tuples $(X_1,\ldots,X_{n-1})$ such that $X_1^2 + \ldots + X_{n-1}^2 = 0$, i.e., we want the sum of $n-1$-squares to be an isotropic quadratic form. This occurs over $\mathbb{Z}_3$ (equivalently, over $\mathbb{Q}_3$) iff $n-1 \geq 3$.