TITLE: Does every polynomial diophantine equation have solutions modulo p?
QUESTION [7 upvotes]: Obviously, this is not exactly true; what I am really asking is whether any diophantine polynomial equation with integer coefficients (let's call them DPEICs) who's solution does not admit contradictory results (eg, x=x+1) has a solution modulo a prime number, and more generally, whether a system of n DPEICs in n variables has a solution (again assuming non-contradictory equations).
More succinctly, if a system of PEICs has a solution in the complex numbers, does it have an integer solution modulo p?

REPLY [2 votes]: Fermat's theorem that $a^p\equiv a\pmod p$ for any inegers $a,p$ with $p$ prime 
 allows us to conclude that the equation $X^p-X+1=0$ for any prime number $p$ does not have any solution modulo that prime number $p$..
In general, as polynomial functions and polynomials are two different animals for finite fields we have many non-constant polynomials that are constant as functions. 
Actually a friend of mine used to joke that $\mathbf{Z}/2$ is an algebraically closed field:
any non-constant polynomial function over $\mathbf{Z}/2$ should assume more than one value hence it has a zero!