MHB What is the name of this theorem in Abstract Algebra

[Amer]

Hi,
There is a theorem in Abstract which said if g.c.d(x,y)= d (g.c.d the greatest common divisor between x and y) then there exist an integers a,b such that

ax + by = d

It is a corollary from Euclidean algorithm.

Does it has a name ?
Thanks in advance.

 

[chisigma]

Hi,
There is a theorem in Abstract which said if g.c.d(x,y)= d (g.c.d the greatest common divisor between x and y) then there exist an integers a,b such that

ax + by = d

It is a corollary from Euclidean algorithm.

Does it has a name ?
Thanks in advance.

In...

Number Theory/Elementary Divisibility - Wikibooks, open books for an open world

... the theorem is indicated simply as 'theorem I'...

Kind regards

$\chi$ $\sigma$

 

[Evgeny.Makarov]

It's called Bézout's identity.

 

[Amer]

Thanks very much, you are great. :D

 

[Deveno]

Not only do the integers $a$ and $b$ exist, but furthermore $d$ is minimal among all POSITIVE $\Bbb Z$-linear combinations of $x$ and $y$.

The reason this is important, is because $a,b$ are in general, not unique, but $d$ is. For example:

gcd(4,6) = 2, and we have:

2 = (1)(6) + (-1)(4) but also:

2 = (-5)(6) + (8)(4) (for example).

The Bezout identity is so useful that integral domains in which it holds are given their own name: Bezout domains (this is a slightly stronger condition than any two elements just having a gcd). These domains are "almost PID's (principal ideal domains)", they share many of the same properties of PID's, but do not have to be Noetherian.

Perhaps this is "too much information". Simpler version: there are many kinds of structures in which SOME of the intuitions we have from integers still apply, but not ALL of them.

 

[Amer]

Not only do the integers $a$ and $b$ exist, but furthermore $d$ is minimal among all POSITIVE $\Bbb Z$-linear combinations of $x$ and $y$.

The reason this is important, is because $a,b$ are in general, not unique, but $d$ is. For example:

gcd(4,6) = 2, and we have:

2 = (1)(6) + (-1)(4) but also:

2 = (-5)(6) + (8)(4) (for example).

The Bezout identity is so useful that integral domains in which it holds are given their own name: Bezout domains (this is a slightly stronger condition than any two elements just having a gcd). These domains are "almost PID's (principal ideal domains)", they share many of the same properties of PID's, but do not have to be Noetherian.

Perhaps this is "too much information". Simpler version: there are many kinds of structures in which SOME of the intuitions we have from integers still apply, but not ALL of them.

Awesome additions, thanks