Euclid's lemma

== English == === Alternative forms === Euclid's Lemma === Etymology === Named after ancient Greek mathematician Euclid of Alexandria (fl. 300 BCE). A version of the proposition appears in Book VII of his Elements. === Noun === Euclid's lemma (uncountable) (number theory) 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;(algebra, by generalisation) 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. ==== 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 === Fundamental theorem of arithmetic on Wikipedia.Wikipedia Principal ideal domain on Wikipedia.Wikipedia Bézout's identity on Wikipedia.Wikipedia Schreier domain on Wikipedia.Wikipedia Euclid's Lemma on Wolfram MathWorld alternative proof of Euclid’s lemma on PlanetMath.org