Irreducible polynomial

An irreducible polynomial is a nonconstant polynomial that cannot be written as a product of two nonconstant polynomials over its coefficient field. Over a general integral domain, neither factor may be a unit.

Let $f\in \mathbb{Z}[X]$ be primitive polynomial and let $p$ be a prime number not dividing its leading coefficient. If its coefficientwise reduction is irreducible in ${\mathbb{F}}_p[X]$, then $f$ is irreducible in $\mathbb{Q}[X]$. Indeed, a rational factorization would give an integer factorization by Gauss's lemma. The leading coefficients of both factors remain nonzero modulo $p$, so reduction preserves their positive degrees, contradicting irreducibility of the reduction. For monic polynomials the leading-coefficient condition is automatic.

If $f$ is an irreducible polynomial over a field $F$, then a nonzero residue $[g]$ modulo $(f)$ has $\gcd (f,g)=1$. The Euclidean algorithm supplies $uf+vg=1$, giving inverse $[v]$ for $[g]$. Thus the quotient ring $F[X]/(f)$ is a field. If $\mathrm{deg}f=d$ and $F$ has $q$ elements, polynomial division supplies exactly $q^d$ residue classes.