Étienne Bézout(1730-1783)

정수론 정리중에 베주 항등식 (Bézout's Identity)라는 유명한 정리가 있음

Proposition: Let a, b be nonzero integers and d be gcd(a, b). Then there are integers x, y such that ax + by = d.

대충 어떤 두 정수의 최대공약수는 원래 정수의 선형결합으로 표현된다는 말임. 

호제법 존나게쓰면 언젠가 저 결과에 다다르긴 하는데 그건 너무 안예쁘니 N의 well-ordering(정렬순서성)을 사용한 커여운 증명을 보는게 정신건강에 좋음

여기서 Well-ordering이란, 

Axiom. (Well-Ordering of N) Every nonempty subset of N has a minimal element.

를 뜻함

[Proof]

Let a, b and d be as above. Define a subset S of N by S = {ax + by > 0 | x, y integers}. Then S is nonempty since it has |a| (with y = 0). Therefore S has a minimum element d = as + bt by the well-ordering of N. We claim d = gcd(a, b).

First, we prove d is a common divisor of a and b. Applying the division algorithm to a, we obtain a = dq + r (0 <= r < d). If r = 0, then d | a. Otherwise, r > 0 and, since r  = a - qd = a - q(as + bt) = a(1 - qs) - bqt, r is contained in S. However, r < d contradicts the minimality of d in S and thus d | a. A similar argument shows d | b and thus d is a common divisor of a and b.

Maximality of d among the common divisors of a and b naturally follows from the fact that d is a linear combination of a and b. Therefore d = gcd(a, b).□

대수쌤이 최소원 가지지고 뭔짓거리하는거 현대대수에서 맨날나온다고 이 증명 외우라고했음. 님들도 외우셈