Quotient by an irreducible polynomial is a field
= Quotient by an irreducible polynomial is a field
{title2=$F[X]/(f)$}
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 $\deg f=d$ and $F$ has $q$ elements, polynomial division supplies exactly $q^d$ residue classes.