What should be assumed about the integral polynomials f1(x),... ,fk(,x) in order that the solvability of the congruence f1(x)f2(x) ... fk(x) is equivalent to0 (mod p) for sufficiently large primes p implies the solvability of the equation f1(x)f2(x) ... fk(x) = 0 in integers x? We provide some explicit characterizations for the cases when fj (x) are binomials or have cyclic splitting fields.
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