Euclid's lemma

English

Alternative forms

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Etymology

Named after ancient Greek mathematician Template:W (fl. 300 BCE). A version of the proposition appears in Book VII of his Template:W.

Noun

1. Template:Lb The proposition that if a prime number p divides an arbitrary product ab of integers, then p divides a or b or both;
   slightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c;
   Template:Lb the proposition that for elements a, b, c of a given principal ideal domain, if a divides bc and gcd(a, b) = 1, then a divides c.
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Usage notes

The proposition as generalised to principal ideal domains is occasionally called Gauss's lemma; some writers, however, consider this usage erroneous as another result is known by that term.

Further reading

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• Euclid's Lemma on Template:W
• alternative proof of Euclid’s lemma on Template:W