Question

证明：如果f(x),g(x)不全为零且u(x)f(x)+v(x)g(x)=(f(x),g(x)),

证明：如果f(x),g(x)不全为零且u(x)f(x)+v(x)g(x)=(f(x),g(x)),那么（u(x),v(x)）=1

Answer 1

令(f(x),g(x))=d(x),f(x)=d(x)r(x),g(x)=d(x)s(x).已知等式两边同除以d(x),得u(x)r(x) v(x)s(x)=1.即证(u(x),v(x))=1.

追答

设：f(x)=f1(x)(f(x),g(x));
g(x)=g1(x)(f(x),g(x));
u(x)f1(x))(f(x),g(x))+v(x)g1(x))(f(x),g(x))=(f(x),g(x)),
由消去律：u(x)f1(x)+v(x)g1(x)=1
所以：(u(x),v(x))=1