On Alternatives of Polynomial Congruences

Mariusz Skałba¹

¹ Institute of Mathematics Polish Academy of Sciences P.O. Box 21 00-956 Warszawa, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 52 (2004) no. 2, pp. 123-132.

Résumé

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.

Détail

DOI : 10.4064/ba52-2-3

Mots-clés : what should assumed about integral polynomials ldots order solvability congruence cdots equiv pmod sufficiently large primes implies solvability equation cdots integers provide explicit characterizations cases binomials have cyclic splitting fields