On Alternatives of Polynomial Congruences
Volume 52 / 2004
Bulletin Polish Acad. Sci. Math. 52 (2004), 123-132
MSC: 11R20, 11A15.
DOI: 10.4064/ba52-2-3
Abstract
What should be assumed about the integral polynomials $f_{1}(x),\ldots,f_{k}(x)$ in order that the solvability of the congruence $f_{1}(x)f_{2}(x)\cdots f_{k}(x)\equiv 0\pmod{p}$ for sufficiently large primes $p$ implies the solvability of the equation $f_{1}(x)f_{2}(x)\cdots f_{k}(x)=0$ in integers $x$? We provide some explicit characterizations for the cases when $f_j(x)$ are binomials or have cyclic splitting fields.
