Euclid's Lemma for Irreducible Elements

Lemma
Let $\left({D, +, \times}\right)$ be a Euclidean domain whose unity is $1$.

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

Let $a, b \in D$ such that:
 * $p \mathop \backslash a \times b$

where $\backslash$ means is a divisor of.

Then $p \mathop \backslash a$ or $p \mathop \backslash b$.

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

Let $n \in D$ such that:
 * $\displaystyle n = \prod_{i \mathop = 1}^r a_i$

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

If $p$ divides $n$, then $p$ divides $a_i$ for some $i$.

That is:
 * $p \mathop \backslash a_1 a_2 \ldots a_n \implies p \mathop \backslash a_1 \lor p \mathop \backslash a_2 \lor \cdots \lor p \mathop \backslash a_n$

Proof
Let $p \mathop \backslash a \times b$.

Suppose $p \nmid a$. Then from the definition of irreducible, $p \perp a$.

Thus from Euclid's Lemma for Euclidean Domains it follows that $p \mathop \backslash b$.

Similarly, if $p \nmid b$ it follows that $p \mathop \backslash a$.

So:
 * $p \mathop \backslash a b \implies p \mathop \backslash a$ or $p \mathop \backslash b$

as we needed to show.

Proof of Generalized Lemma
Proof by induction:

For all $r \in \N_{>0}$, let $P \left({r}\right)$ be the proposition:
 * $\displaystyle p \mathop \backslash \prod_{i \mathop = 1}^r a_i \implies \exists i \in \left[{1 \,.\,.\, r}\right]: p \mathop \backslash a_i$

$P (1)$ is true, as this just says $p \mathop \backslash a_1 \implies p \mathop \backslash a_1$.

Basis for the Induction
$P (2)$ is the case:
 * $p \mathop \backslash a_1 a_2 \implies p \mathop \backslash a_2$ or $p \mathop \backslash a_2$

which is the main lemma as proved above.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:


 * $\displaystyle p \mathop \backslash \prod_{i \mathop = 1}^k a_i \implies \exists i \in \left[{1 \,.\,.\, k}\right]: p \mathop \backslash a_i$

Then we need to show:


 * $\displaystyle p \mathop \backslash \prod_{i \mathop = 1}^{k+1} a_i \implies \exists i \in \left[{1 \,.\,.\, {k+1}}\right]: p \mathop \backslash a_i$

Induction Step
This is our induction step:

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \forall r \in \N: p \mathop \backslash \prod_{i \mathop = 1}^r a_i \implies \exists i \in \left[{1 \,.\,.\, r}\right]: p \mathop \backslash a_i$

Also see

 * Euclid's Lemma for Prime Divisors, for the usual statement of this result, which is this lemma as applied specifically to the integers.