Euclid's Lemma for Unique Factorization Domain/General Result

Lemma

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

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

Identical to the proof of Euclid's Lemma for Irreducible Elements: General Result.

$\blacksquare$

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Source of Name

This entry was named for Euclid.