Greatest Common Divisor of Polynomials

Definition

• Let $p_1,\dots ,p_n\in \mathbb{F}[x]$
• A Monic Polynomial $d\in \mathbb{F}[x]$ such that:
  1. $d\in \mathbb{F}[x]\{p_1,\dots ,p_n\}$ ( $=\mathbb{F}[x]p_1+\cdots +\mathbb{F}[x]p_n$)
  2. $d$ divides $p_i,\forall i$
  3. $d$ divides every polynomial that divides $p_i,\forall i$ (this divisor is the greatest)

We can say that $p_1,\dots ,p_n$ is Relatively Prime if $gcd(p_1,\dots ,p_n)=1$