
theorem mm6b:
for F being 0-characteristic Field
for p being Element of the carrier of Polynom-Ring F
holds (Deriv F).p = 0_.(F) iff p is constant
proof
let F be 0-characteristic Field;
let p be Element of the carrier of Polynom-Ring F;
now assume (Deriv F).p = 0_.(F); then
  A1: deg (Deriv F).p = -1 by HURWITZ:20;
  per cases;
  suppose p is zero;
    hence p is constant;
    end;
  suppose p is non zero; then
    deg (Deriv F).p = deg p - 1 by mm6c;
    hence p is constant by A1,RING_4:def 4;
    end;
  end;
hence thesis by der4;
end;
