
theorem Th48:
  cyclotomic_poly(1) = <%-1_F_Complex, 1_F_Complex %>
proof
  set cMGFC = the carrier of MultGroup F_Complex;
  consider s being non empty finite Subset of F_Complex such that
A1: s = { y where y is Element of cMGFC : ord y = 1 } and
A2: cyclotomic_poly(1) = poly_with_roots((s,1)-bag) by Def5;
A3: 1-roots_of_1 = {x where x is Element of cMGFC : ord x divides 1} by Th35;
  now
    let x be object;
    hereby
      assume x in s;
      then ex x1 being Element of cMGFC st x = x1 & ord x1 = 1 by A1;
      hence x in 1-roots_of_1 by A3;
    end;
    assume x in 1-roots_of_1;
    then consider x1 being Element of cMGFC such that
A4: x = x1 and
A5: ord x1 divides 1 by A3;
    ord x1 = 1 by A5,WSIERP_1:15;
    hence x in s by A1,A4;
  end;
  then s = 1-roots_of_1 by TARSKI:2;
  hence thesis by A2,Lm4,Th46;
end;
