Let $S=\mathbb F_q[x]$ be the polynomial ring over the finite field $\mathbb F_q$. If $I=\langle p(x)\rangle$ is a maximal ideal of $S$ ($p(x)$ is irreducible), then the field $S/I$ is also a finite field. This is a basic result of field theory.
Now, consider the polynomial ring in $n$ indeterminates $R=\mathbb F_q[x_1, \dots , x_n]$. Let $J=\langle f_1(x_1, ..., x_n), \dots, f_n(x_1, ..., x_n)\rangle $ be a maximal ideal of $R$ (the polynomials $f_i$, for $i=1, ..., n$, are irreducible). It is true that in this case the field $R/J$ is also finite?