 Polynomial congruence

A polynomial congruence asks for integer residue classes satisfying an equation such asInteger coefficients ensure that replacing $x$ by a congruent integer changes $f(x)$ by a multiple of $m$. The problem is therefore an equation in the quotient ring $\mathbb{Z}/m\mathbb{Z}$, rather than infinitely many unrelated integer tests.

For $x^2\equiv 1(\mathrm{mod}8)$, the solutions are $1,3,5,7$ modulo eight. This quadratic has four roots because the quotient ring has zero divisors: $(x-1)(x+1)=0$ does not force either factor to vanish individually. Over a prime modulus, a nonzero degree-$d$ polynomial has at most $d$ roots because the quotient is a field. The nonzero-polynomial qualification matters if all coefficients vanish modulo the prime.

For coprime moduli, the Chinese remainder theorem combines solutions componentwise. Thus solving modulo prime powers and then recombining is a natural strategy; lifting between powers of one prime introduces additional conditions, as in Hensel's lemma.