Question

对于F【X】中的多项式f(x),g(x).若(f(x),f(x)+g(x))=1,求证(f(x),g(x))=1

Answer 1

证明：(f(x),g(x))=1,则存在a(x),b(x)使得af + bg = 1； 假设t(x)=(g(x),h(x)),则存在c(x),d(x)使得cg + dh = t 则t = cg + dh * 1 = cg + dh * (af + bg ) = cg + adfh + bdhg = (ad)fh + (c+ bdh)g 从而存在u(x) = a(x)d(x),v(x) = c(x) + b(x)d(x)h(x),使得ufh + vg = t,而t|h,所以t|fh,t|g,故t是fh和g的最大公因式.从而(f(x)h(x),g(x))=(g(x),h(x)) 得证.