Bézout's Identity/Principal Ideal Domain

Theorem

Let $\struct {D, +, \circ}$ be a principal ideal domain.

Let $S = \set {a_1, a_2, \dotsc, a_n}$ be a set of non-zero elements of $D$.

Let $y$ be a greatest common divisor of $S$.

Then $y$ is expressible in the form:

$y = d_1 a_1 + d_2 a_2 + \dotsb + d_n a_n$

where $d_1, d_2, \dotsc, d_n \in D$.

Proof

From Finite Set of Elements in Principal Ideal Domain has GCD we have that at least one such greatest common divisor exists.

So, let $y$ be a greatest common divisor of $S$.

Let $J$ be the set of all linear combinations in $D$ of $\set {a_1, a_2, \dotsc, a_n}$.

From Set of Linear Combinations of Finite Set of Elements of Principal Ideal Domain is Principal Ideal:

$J = \ideal x$

for some $x \in D$, where $\ideal x$ denotes the principal ideal generated by $x$.

From Finite Set of Elements in Principal Ideal Domain has GCD, $x$ is a greatest common divisor of $S$.

From Greatest Common Divisors in Principal Ideal Domain are Associates, $y$ is an associate of $x$.

By definition of associate:

$\ideal y = \ideal x$

Therefore:

$y \in J$

and so by definition, $y$ is expressible in the form:

$y = d_1 a_1 + d_2 a_2 + \dotsb + d_n a_n$

where $d_1, d_2, \dotsc, d_n \in D$.

$\blacksquare$

Also known as

Bézout's Identity is also known as Bézout's lemma, but that result is usually applied to a similar theorem on polynomials.

Some sources omit the accent off the name: Bezout's identity (or Bezout's lemma), which may be a mistake.

Source of Name

This entry was named for Étienne Bézout.

Historical Note

There are sources which suggest that Bézout's Identity was first noticed by Claude Gaspard Bachet de Méziriac.

Étienne Bézout's contribution was to prove a more general result, for polynomials.

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 62.4$ Factorization in an integral domain: $\text{(iii)}$