2.互素的判定与性质定理3f(x),g(x)E P[xl, f(x),g(x) 互素台3u(x),v(x)E P[x], 使u(x)f(x)+v(x)g(x) =1证:"→"显然。""设(x)为f(x),g(x)的任一公因式,则p(x)[f(x), (x)g(x), 从而 p(x)1, 又 1(x),:: p(x)=c, c0. 故 (f(x),g(x)=1.81.4最大公因式R下§1.4 最大公因式 定理3 互素 ,使 f x g x P x ( ), ( ) [ ] , f x g x ( ), ( ) u x f x v x g x ( ) ( ) ( ) ( ) 1 + = u x v x P x ( ), ( ) [ ] 2.互素的判定与性质 证: " " 显然. " " 设 ( ) ( ), ( ) x f x g x 为 的任一公因式,则 ( ) ( ), ( ) ( ), x f x x g x 从而 ( ) 1, x 又 1 ( ), x = ( ) , 0. x c c 故 ( ( ), ( )) 1. f x g x =