Consider the n-variable polynomial ring over $F_2$, denoted $F_2[x_1, x_2, x_3, ..., x_n]$. Some polynomials in this ring have no roots in F_2.
To construct roots, we can create the algebraic closure $\overline{F_2}$. This adds roots to those polynomials that would otherwise have none.
My question: given a polynomial in $F_2[x_1, x_2, x_3, ..., x_n]$ that already has a root in $F_2$, when can $\overline{F_2}$ add additional roots?
In other words, is there something like a fundamental theorem of algebra, but for multivariate polynomials over finite fields?