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)

1. (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.
   • 1998, Peter M. Higgins, Mathematics for the Curious, Oxford University Press, page 78:

     I used Euclid's Lemma in a slightly sly way in the second chapter, where I ran through the argument that ${\sqrt {2}}$ is irrational. I said there that if $2$ is a factor of $a^{2}$ then $a$ itself must be even. This follows from Euclid's Lemma upon taking $p=2$, the only even prime, and taking $b=a$. Indeed, using Euclid's Lemma it is not hard to generalize the argument showing ${\sqrt {2}}$ to be irrational to prove that ${\sqrt {p}}$ is irrational for any prime $p$.

   • 2007, David M. Burton, The History of Mathematics, McGraw-Hill, page 179:

     If $a$ and $b$ are not relatively prime, then the conclusion of Euclid's lemma may fail to hold. A specific example: $12\mid 9\cdot 8$, but $12\nmid 9$ and $12\nmid 8$.

   • 2008, Martin Erickson, Anthony Vazzana, Introduction to Number Theory‎^[1], Taylor & Francis (Chapman & Hall / CRC Press), page 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 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