# Euclid's Lemma for Irreducible Elements/General Result

## Contents

## Lemma

Let $\struct {D, +, \times}$ be a Euclidean domain whose unity is $1$.

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:

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

## Proof

Proof by induction:

For all $r \in \N_{>0}$, let $\map P r$ be the proposition:

- $\displaystyle p \divides \prod_{i \mathop = 1}^r a_i \implies \exists i \in \closedint 1 r: p \divides a_i$

$\map P 1$ is true, as this just says $p \divides a_1 \implies p \divides a_1$.

### Basis for the Induction

$\map P 2$ is the case:

- $p \divides a_1 a_2 \implies p \divides a_2$ or $p \divides a_2$

which is proved in Euclid's Lemma for Irreducible Elements.

This is our basis for the induction.

### Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:

- $\displaystyle p \divides \prod_{i \mathop = 1}^k a_i \implies \exists i \in \closedint 1 k: p \divides a_i$

Then we need to show:

- $\displaystyle p \divides \prod_{i \mathop = 1}^{k + 1} a_i \implies \exists i \in \closedint 1 {k + 1}: p \divides a_i$

### Induction Step

This is our induction step:

\(\displaystyle p\) | \(\divides\) | \(\displaystyle a_1 a_2 \ldots a_{k + 1}\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle p\) | \(\divides\) | \(\displaystyle \paren {a_1 a_2 \ldots a_k} \paren {a_{k + 1} }\) | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle p\) | \(\divides\) | \(\displaystyle a_1 a_2 \ldots a_k \lor p \divides a_{k + 1}\) | Basis for the Induction | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle p\) | \(\divides\) | \(\displaystyle a_1 \lor p \divides a_2 \lor \ldots \lor p \divides a_k \lor p \divides a_{k + 1}\) | Induction Hypothesis |

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

Therefore:

- $\displaystyle \forall r \in \N: p \divides \prod_{i \mathop = 1}^r a_i \implies \exists i \in \closedint 1 r: p \divides a_i$

$\blacksquare$

## Source of Name

This entry was named for Euclid.

## 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.

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 29$. Irreducible elements: Theorem $56 \ \text{(ii)}$