Euclid's Lemma for Unique Factorization Domain

Lemma

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

Let $p$ be an irreducible element of $D$.

Let $a, b \in D$ such that:

$p \divides a \times b$

where $\divides$ means is a divisor of.

Then $p \divides a$ or $p \divides b$.

General Result

Let $p$ be an irreducible element of $D$.

Let $n \in D$ such that:

$\ds n = \prod_{i \mathop = 1}^r a_i$

where $a_i \in D$ for all $i: 1 \le i \le r$.

Then if $p$ divides $n$, it follows that $p$ divides $a_i$ for some $i$.

That is:

$p \divides a_1 a_2 \ldots a_n \implies p \divides a_1 \lor p \divides a_2 \lor \cdots \lor p \divides a_n$

Proof

The elements $a$ and $b$ have complete factorizations:

$a = u \times x_1 \times x_2 \times \cdots \times x_n$

and

$b = v \times y_1 \times y_2 \times \cdots \times y_m$

where $u$ and $v$ are units of $D$ and $x_1, x_2, \ldots, x_n$ and $y_1, y_2, \ldots, y_m$ are irreducible elements.

Then the unique complete factorization of $a \times b$ into irreducible elements is:

$(1): \quad a \times b = \paren {u \times v} \times x_1 \times x_2 \times \cdots \times x_n \times y_1 \times y_2 \times \cdots \times y_m$

If $p$ is an associate of $x_i$ for some $i$ then we will have $p \divides a$.

Similarly, if $p$ is an associate of $y_i$ for some $i$ then $p \divides b$.

Aiming for a contradiction, suppose that $p$ is not an associate of $x_i$ or $y_i$ for all $i$.

Since $p \divides a \times b$, there is a $c \in D$ such that $p \times c = a \times b$.

This element $c$ has the unique complete factorization:

$c = w \times z_1 \times z_2 \times \cdots \times z_l$

where $w$ is a unit of $D$ and $z_1, z_2, \ldots, z_l$ are irreducible elements.

This gives us another complete factorization:

$a \times b = w \times p \times z_1 \times z_2 \times \cdots \times z_l$

But the complete factorization $(1)$ contains no irreducible element that is an associate of $p$.

This contradicts the unique factorization of $a \times b$.

We conclude that $p$ is an associate of $x_i$ or $y_i$ for some $i$.

Therefore, $p \divides a$ or $p \divides b$.

$\blacksquare$

Source of Name

This entry was named for Euclid.