Euclid's lemma

English Alternative forms en Euclid's Lemma Etymology Named after ancient Greek mathematician Euclid Euclid of Alexandria (fl. 300 BCE). A version of the proposition appears in Book VII of his Euclids Elements Elements. Noun - en 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;en 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. en 1998 Peter M. Higgins Mathematics for the Curious https://books.google.com.au/books?id=SLqu0a_6uzUC&pg=PA78&dq=%22Euclid%27s+lemma%22&hl=en&sa=X&ved=2ahUKEwj9vYKztaTrAhUDX30KHecMD7cQ6AEwBHoECCMQAg#v=onepage&q=%22Euclid's%20lemma%22&f=false 78 w:Oxford University Press I used Euclid's Lemma in a slightly sly way in the second chapter, where I ran through the argument that is irrational. I said there that if is a factor of then itself must be even. This follows from Euclid's Lemma upon taking , the only even prime, and taking . Indeed, using Euclid's Lemma it is not hard to generalize the argument showing to be irrational to prove that is irrational for any prime . en 2007 David M. Burton The History of Mathematics McGraw-Hill https://books.google.com.au/books?id=VDlGAQAAIAAJ&q=%22Euclid%27s+lemma%22&dq=%22Euclid%27s+lemma%22&hl=en&sa=X&ved=2ahUKEwj9vYKztaTrAhUDX30KHecMD7cQ6AEwDnoECD4QAg 179 If and are not relatively prime, then the conclusion of Euclid's lemma may fail to hold. A specific example: , but and . en 2008 Martin Erickson; Anthony Vazzana Introduction to Number Theory Taylor & Francis (Chapman & Hall / CRC Press) https://books.google.com.au/books?id=BeFIIvcBT80C&pg=PA42&dq=%22Euclid%27s+lemma%22&hl=en&sa=X&ved=2ahUKEwj9vYKztaTrAhUDX30KHecMD7cQ6AEwB3oECDIQAg#v=onepage&q=%22Euclid's%20lemma%22&f=false 42 In our discussion of Euclid's lemma (Corollary 2.18), we noted that the uniqueness of factorization of integers is a fact that we often take for granted given the way it is introduced in school. 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 Principal ideal domain Bézout's identity Schreier domain Euclid's Lemma on MathWorld Wolfram MathWorld alternative proof of Euclid’s lemma on PlanetMath PlanetMath.org