A polynomial reduced modulo a prime number $p$ is a polynomial $p(x)=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n-1}x^{n-1}+a_nx^{n},$ over the ring of integers $\mathbb Z[X],$ in which all coefficients have been replaced by the congruence classes modulo a prime number $q$, i.e. the polynomial
$$p(x) \mod q:=a_{0}(p)+a_{1}x+a_{2}x^{2}+\cdots+a_{n-1}x^{n-1}+a_nx^{n}\mod q.$$
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