English Dictionary

Euclid's lemma

noun

(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;

Euclid's lemma is a fundamental result in number theory.

意味（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;

I used Euclid's Lemma in a slightly sly way in the second chapter, where I ran through the argument that √2 is irrational. I said there that if 2 is a factor of a² 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 √2 to be irrational to prove that √p is irrational for any prime p.

意味（2）

slightly more generally, the proposition that for integers a, b, c, if a divides bc and gcd(a, b) = 1, then a divides c;

意味（3）

(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.