
theorem
X^3-1 = X-(1.F_Complex) * X-zeta * X-(zeta^2)
proof
reconsider q = X^2+X+1 as Polynomial of F_Rat;
reconsider p = X-zeta * X-(zeta^2) as Polynomial of F_Complex
  by POLYNOM3:def 10;
B: X-(1.F_Complex) = rpoly(1,1.F_Complex) by FIELD_9:def 2;
C: X-(1.F_Rat) = rpoly(1,1.F_Rat) by FIELD_9:def 2;
   1.F_Rat = 1.F_Complex by GAUSSINT:13,COMPLEX1:def 4,COMPLFLD:def 1; then
A: X-(1.F_Complex) = X-(1.F_Rat) by B,C,FIELD_4:21;
thus X^3-1 = rpoly(1,1.F_Rat) *' q by C,lemX3,POLYNOM3:def 10
     .= rpoly(1,1.F_Complex) *' p by A,B,C,FIELD_4:17,lemX3a
     .= X-(1.F_Complex) * (X-zeta * X-(zeta^2)) by B,POLYNOM3:def 10
     .= X-(1.F_Complex) * X-zeta * X-(zeta^2) by GROUP_1:def 3;
end;
