I am working on a problem, and I am not understanding the language very well. Here is the setup of the problem:
Consider the set $\{ 0, 1, 2 \}$ with the operations addition $(+)$ modulo $3$ and multiplication $()$ modulo $3$. Let $\mathbb{Z}_3$ denote the resulting field structure.
An algebraic expression over $\mathbb{Z}_3$ using variables $X_1, X_2, \ldots, X_n$, operations $\{ +, -, * \}$ and the constants $0$ and $1$ is a multi-variate polynomial using these variables and constants with all monomials having coefficient $1$.
Example: $X_1^3*X_2 - X_1^2*X_3 + X_2^4*X_3^2 + 1$
Show that the following problem is NP-complete:
INPUT: An algebraic expression $P$ over $\mathbb{Z}_3$ with variables $X_1, X_2, \ldots,X_n$.
PROBLEM: Is there an assignment of values from $\mathbb{Z}_3$ to the variables so that the expression $P$ evaluates to a non-zero value? (Hint: $a^2 = 1$ for all non-zero a over $\mathbb{Z}_3$.)
I have a field structure set up (I think). It's just a big table with the set $\{ 0, 1, 2 \}$ on the top and left side where the innards are just either addition $\mod 3$ or multiplication $\mod 3$ of the row and column. So, if that is $\mathbb{Z}_3$, where do I go from there to set up the algebraic expression that I need to solve this problem?