> [!NOTE] Definition 1 (Relatively Prime Polynomials) > Let $(D,+,\times)$ be an [[Integral Domain|integral domain]]. Let $D[x]$ be the [[Ring of Polynomial Forms|ring of polynomial forms]] over $D$ in $x.$ Let $f,g\in F[x].$ Then $f$ and $g$ are *relatively prime* iff any common [[Divisibility in Ring of Polynomial Forms|factor]] of $f$ and $g$ is a [[Units of Ring of Polynomial Forms over Integral Domain|unit]]: for all $h \in F[x] \setminus \{ 0 \}:h\mid f \land h \mid g \implies \deg(h)=0.$ > [!NOTE] Definition 2 (Relatively Prime Polynomials) > Let $f,g \in D[x].$ Then $f$ and $g$ are relatively prime iff any [[Greatest Common Divisor of Polynomial Forms Over Integral Domain|gcd]] of $f$ and $g$ is a unit. # Properties By [[Bézout's Identity for Ring of Polynomial Forms Over Field|Bézout's identity]], if $f,g$ are relatively prime polynomials over a field, then there exists $x,y\in D[x]$ such that $fx+gy=1.$