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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?

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  • $\begingroup$ Answer is always if $n > 1$. Over $\mathbb F_2$ every polynomial has only finitely many zeroes (as $\mathbb F_2^n$ is finite itself), but over algebraic closure, which is infinite, constituting some elements of $\overline{\mathbb F_2}$ in last $n-1$ variable gives you polynomial in 1 variable which evidently has roots! (that's the definition of algebraic closure, after all). So you have infinitely many disjoint lines, and polynomial vanishes at some point of each one. $\endgroup$ – xsnl Apr 3 '17 at 15:29
  • $\begingroup$ Much harder problem is to determine exact speed at which zero set grows as you increase size of your field — not to algebraic closure, but to some finite extension. en.wikipedia.org/wiki/Weil_conjectures $\endgroup$ – xsnl Apr 3 '17 at 15:33

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