Bezout's Theorem

Definition

In integral domain $D$, the following equation is called the Bézout’s identity. $$ m a + n b = \gcd \left( a, b \right) $$

If there exists $m,n \in D$ satisfying the Bézout’s identity for all $a, b \in D$, then $D$ is called a Bézout Domain.

$\gcd (a,b)$ is the greatest common divisor of $a,b$.

Theorem

PID is a Bézout Domain

A Principal Ideal Domain is a Bézout Domain. In other words, for every $a, b \in R$ in Principal Ideal Domain $R$, there always exists $m,n \in R$ that satisfies the Bézout’s identity. $$ m a + n b = \gcd \left( a, b \right) $$

Proof ¹

Let’s assume $a ,b \in R$ as $d := \gcd \left( a,b \right)$. Since $R$ is a PID, $a R + b R$ is also a PID, $$ a R + b R = c R $$ and there exists $c \in R$ that satisfies it. Since $d$ is the least common multiple of $a,b$, it is $a R + b R \subset d R$, and consequently, $$ a R + b R \subset d R \subset c R = a R + b R $$ holds. As the leftmost side and the rightmost side are equal, it is $a R + b R = d R$, and the following $m , n \in R$ must exist. $$ ma + nb = d $$

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Explanation

Can be seen as a generalization of the Extended Euclidean Theorem that appears in number theory.

Note that for the converse of the theorem, for a Bézout Domain to be a PID, it also needs to be a Unique Factorization Domain.²