Elements of Euclidean Domain have Greatest Common Divisor

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Theorem

Let $\struct {D, +, \times}$ be a Euclidean domain.

Then any two elements $a, b \in D$ have a greatest common divisor $d$ such that:

$d \divides a \land d \divides b$
$x \divides a \land x \divides b \implies x \divides d$

and $d$ is written $\gcd \set {a, b}$.

For any $a, b \in D$:

$\exists s, t \in D: s a + t b = d$

Any two greatest common divisors of any $a, b$ are associates.


Proof

Let $a, b \in D$.

Let $U \subseteq D$ be the set of all elements $h a + k b$ of $D$ where $h, k \in D$.

Then $U$ is an ideal of $D$.

Note that $U = \ideal a + \ideal b$ where $\ideal a$ and $\ideal b$ are Principal Ideal.

By Sum of Ideals is Ideal, $U$ is an ideal.

By Euclidean Domain is Principal Ideal Domain, $U$ is a principal ideal, $\ideal d$ say.

As $a, b \in U$ it follows that $d$ is a divisor of $a$ and $b$, that is:

$d \divides a \land d \divides b$

Since $d$ itself is in $U$, we have:

$\exists s, t \in D: s a + t b = d$

By Common Divisor in Integral Domain Divides Linear Combination:

$x \divides a \land x \divides b \implies x \divides d$

So $d$ is a greatest common divisor of $a$ and $b$.


If $d$ and $d'$ are both greatest common divisors of $a$ and $b$, then $d \divides a \land d \divides b$ and so $d \divides d'$.

Similarly $d' \divides d$.

So $d$ and $d'$ are associates.

$\blacksquare$


Sources