Euclid's Lemma for Irreducible Elements

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Lemma

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

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:

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

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

Let $p \divides a \times b$.


Suppose $p \nmid a$.

Then from the definition of irreducible:

$p \perp a$

Thus from Euclid's Lemma for Euclidean Domains:

$p \divides b$

Similarly, if $p \nmid b$:

$p \divides a$

So:

$p \divides a b \implies p \divides a$

or:

$p \divides b$

as we needed to show.

$\blacksquare$


Source of Name

This entry was named for Euclid.


Also see


Sources