f(x)u(x)+g(x)v(x)=d(x)且d(x)|f(x),d(x)|g(x),证明d(x)是f(x),g(x)的最大公因式

题目

f(x)u(x)+g(x)v(x)=d(x)且d(x)|f(x),d(x)|g(x),证明d(x)是f(x),g(x)的最大公因式
高等代数,不用辗转相除法

答案

设h(x)为f(x)和g(x)的最大公因式则
h(x)|f(x),h(x)|g(x)
∴ h(x)|[f(x)u(x)+g(x)v(x)]
又 f(x)u(x)+g(x)v(x)=d(x)
∴ h(x)|d(x) ①
∵ d(x)|f(x),d(x)|g(x)
∴ d(x)是f(x)、g(x)的公因式
∴ d(x)|h(x) ②
由①②得
d(x)=c·h(x) 为f(x)和g(x)的最大公因式,这里c为常数.