
theorem
X^3-2 = X-(3-CRoot(2)) * X-(3-CRoot(2) * zeta) * X-(3-CRoot(2) * zeta^2)
proof
set F = F_Complex;
set a = 3-CRoot(2) * zeta, b = 3-CRoot(2) * zeta^2, c = 3-CRoot(2);
A: X-a = rpoly(1,a) by FIELD_9:def 2 .= <%-a, 1.F%> by RING_5:10;
B: X-b = rpoly(1,b) by FIELD_9: def 2 .= <%-b, 1.F%> by RING_5:10;
C: X-c = rpoly(1,c) by FIELD_9: def 2 .= <%-c, 1.F%> by RING_5:10;
D: (X-a) * (X-b) = <%-a, 1.F%> *' <%-b, 1.F%> by A,B,POLYNOM3:def 10
     .= <%(-a)*(-b),(1.F)*(-b)+(1.F)*(-a),(1.F)*(1.F)%> by FIELD_9:24
     .= <%a*b,-b+-a,1.F%> by VECTSP_1:10;
    F_Real is Subfield of F by FIELD_4:7; then
H0: F_Real is Subring of F by FIELD_5:12;
H1: a * b
     = (3-CRoot(2) * 3-CRoot(2)) * (zeta * zeta^2)
    .= (3-CRoot(2) * 3-CRoot(2)) * zeta|^3 by lemI
    .= 3-Root(2)|^1 * 3-Root(2) by LZ23,BINOM:8
    .= 3-Root(2)|^1 * 3-Root(2)|^1 by BINOM:8
    .= 3-Root(2)|^(1+1) by BINOM:10;
H3: -c = -3-Root(2) by H0,FIELD_6:17;
H4:  1 = 1.F by COMPLFLD:def 1,COMPLEX1:def 4; then
H5: -2 = -(1.F + 1.F) by COMPLFLD:2;
    1 = 1.F by COMPLEX1:def 4,COMPLFLD:def 1; then
H6: -1 = -1.F by COMPLFLD:2;
    zeta^2 + zeta = zeta|^(1+1) + zeta by lemI; then
  b + a
     = 3-CRoot(2) * (-1.F + (-zeta + zeta)) by H6,LZ23
    .= 3-CRoot(2) * (-1.F + 0.F) by RLVECT_1:5
    .= -(3-CRoot(2) * 1.F) by VECTSP_1:8
    .= -3-Root(2) by H0,FIELD_6:17; then
H6: --3-Root(2) = -(b+a) by COMPLFLD:2 .= -b + -a by RLVECT_1:31;
reconsider p1 = X-c as Polynomial of F;
p1 *' <%a*b,-b+-a,1.F%> = X^3-2
  proof
  J: 5 -' 1 = 5 - 1 by XREAL_0:def 2;
  L: 4 -' 1 = 4 - 1 by XREAL_0:def 2; then
  K: 2 -' 1 = 2 - 1 & 3 -' 1 = 3 - 1 & 3 -' 2 = 3 - 2 & 4 -' 1 = 3 &
     4 -' 2 = 4 - 2 by XREAL_0:def 2;
  the carrier of Polynom-Ring F_Rat c=
           the carrier of Polynom-Ring F by FIELD_4:10; then
  X^3-2 is Element of the carrier of Polynom-Ring F; then
  reconsider q = X^3-2 as Polynomial of F;
  set p2 = <%a*b,-b+-a,1.F%>;
  A: dom(p1*'p2) = NAT by FUNCT_2:def 1 .= dom q by FUNCT_2:def 1;
  now let o be object;
  assume o in dom q;
  then reconsider i = o as Element of NAT;
  consider r being FinSequence of the carrier of F such that
  B1: len r = i+1 & (p1*'p2).i = Sum r &
      for k being Element of NAT st k in dom r
      holds r.k = p1.(k-'1) * p2.(i+1-'k) by POLYNOM3:def 9;
  i <= 3 implies i = 0 or ... or i = 3; then
  per cases;
  suppose C1: i = 0;
    then B2: r = <*r.1*> by B1,FINSEQ_1:40;
    then dom r = {1} by FINSEQ_1:2,FINSEQ_1:38;
    then 1 in dom r by TARSKI:def 1;
    then r.1 = p1.(1-'1) * p2.(0+1-'1) by C1,B1
            .= p1.(1-'1) * p2.0 by NAT_2:8
            .= p1.0 * p2.0 by NAT_2:8
            .= p1.0 * (a * b) by FIELD_9:16
            .= (-c) * (a * b) by C,POLYNOM5:38
            .= -(3-Root(2) * 3-Root(2)|^2) by H1,H3
            .= -(3-Root(2)|^2 * 3-Root(2)|^1) by BINOM:8
            .= -(3-Root(2)|^(2+1)) by BINOM:10
            .= -(1.F + 1.F) by H4,COMPLFLD:2,R32;
    hence (p1*'p2).o = q.o by H5,C1,LL0,B1,B2,RLVECT_1:44;
    end;
  suppose C1: i = 1;
    then B3: r = <*r.1,r.2*> by B1,FINSEQ_1:44;
    B4: dom r = {1,2} by B1,C1,FINSEQ_1:def 3,FINSEQ_1:2;
    then 1 in dom r by TARSKI:def 2;
    then B5: r.1 = p1.(1-'1) * p2.(1+1-'1) by C1,B1
            .= p1.0 * p2.1 by K,NAT_2:8
            .= (-c) * p2.1 by C,POLYNOM5:38
            .= (-c) * (-b+-a) by FIELD_9:16;
    2 in dom r by B4,TARSKI:def 2;
    then r.2 = p1.(2-'1) * p2.(1+1-'2) by C1,B1
            .= p1.1 * p2.0 by K,NAT_2:8
            .= 1.F * p2.0 by C,POLYNOM5:38
            .= 1.F * (a * b) by FIELD_9:16;
    then Sum r = (-c) * (-b+-a) + (a * b) by B3,B5,RLVECT_1:45
              .= -(3-Root(2) * 3-Root(2)) + 3-Root(2)|^2 by H3,H6,H1
              .= -(3-Root(2)|^1 * 3-Root(2)) + 3-Root(2)|^2 by BINOM:8
              .= -(3-Root(2)|^1 * 3-Root(2)|^1) + 3-Root(2)|^2 by BINOM:8
              .= -3-Root(2)|^(1+1) + 3-Root(2)|^2 by BINOM:10
              .= 0;
    hence (p1*'p2).o = q.o by B1,C1,LL0;
    end;
  suppose C1: i = 2;
    then B3: r = <*r.1,r.2,r.3*> by B1,FINSEQ_1:45;
    B4: dom r = Seg 3 by B1,C1,FINSEQ_1:def 3
             .= Seg 2 \/ {2+1} by FINSEQ_1:9
             .= {1,2,3} by FINSEQ_1:2,ENUMSET1:3;
    then 1 in dom r by ENUMSET1:def 1;
    then B5: r.1 = p1.(1-'1) * p2.(2+1-'1) by C1,B1
            .= p1.0 * p2.2 by K,NAT_2:8
            .= p1.0 * 1.F by FIELD_9:16
            .= -c by C,POLYNOM5:38;
    2 in dom r by B4,ENUMSET1:def 1;
    then B6: r.2 = p1.1 * p2.1 by K,C1,B1
            .= 1.F * p2.1 by C,POLYNOM5:38
            .= -b+-a by FIELD_9:16;
    3 in dom r by B4,ENUMSET1:def 1;
    then r.3 = p1.2 * p2.(2+1-'3) by K,C1,B1
            .= 0.F * p2.(2+1-'3) by C,POLYNOM5:38;
    then Sum r = -c + (-b + -a) + 0.F by B3,B5,B6,RLVECT_1:46
              .= 0 by H3,H6;
    hence (p1*'p2).o = q.o by B1,C1,LL0;
    end;
  suppose C1: i = 3;
    then B3: r = <*r.1,r.2,r.3,r.4*> by B1,FINSEQ_4:76;
    B4: dom r = Seg 4 by B1,C1,FINSEQ_1:def 3
             .= Seg 3 \/ {3+1} by FINSEQ_1:9
             .= (Seg 2 \/ {2+1}) \/ {4} by FINSEQ_1:9
             .= {1,2,3} \/ {4} by FINSEQ_1:2,ENUMSET1:3
             .= {1,2,3,4} by ENUMSET1:6;
    then 1 in dom r by ENUMSET1:def 2;
    then B5: r.1 = p1.(1-'1) * p2.(3+1-'1) by C1,B1
            .= p1.0 * p2.3 by L,NAT_2:8
            .= p1.0 * 0.F by FIELD_9:16;
    2 in dom r by B4,ENUMSET1:def 2;
    then B6: r.2 = p1.1 * p2.2 by K,C1,B1
            .= p1.1 * 1.F by FIELD_9:16
            .= 1.F by C,POLYNOM5:38;
    3 in dom r by B4,ENUMSET1:def 2;
    then B7: r.3 = p1.2 * p2.(3+1-'3) by K,C1,B1
            .= 0.F * p2.(3+1-'3) by C,POLYNOM5:38;
    4 in dom r by B4,ENUMSET1:def 2;
    then r.4 = p1.3 * p2.(3+1-'4) by L,C1,B1
            .= 0.F * p2.(3+1-'4) by C,POLYNOM5:38;
    then r = <*(0.F),(1.F),(0.F)*> ^ <*(0.F)*> by B3,B5,B6,B7,FINSEQ_4:74;
    then Sum r
       = Sum <*(0.F),(1.F),(0.F)*> + Sum <*(0.F)*> by RLVECT_1:41
      .= Sum <*(0.F),(1.F),(0.F)*> + 0.F by RLVECT_1:44
      .= 0.F + 1.F + 0.F by RLVECT_1:46
      .= 1 by COMPLFLD:def 1,COMPLEX1:def 4;
    hence (p1*'p2).o = q.o by B1,C1,LL0;
    end;
  suppose C11: i > 3;
    reconsider q1 = q as Element of the carrier of Polynom-Ring F
        by POLYNOM3:def 10;
    CC: deg q1 = 3 by LL,FIELD_4:20;
    C0: deg q = len q - 1 by HURWITZ:def 2;
    C1: i >= 3 + 1 by C11,NAT_1:13;
    then E: q.i = 0.F by CC,C0,ALGSEQ_1:8;
    1.F <> 0.F; then
    C2: len p1 = 2 & deg p2 = 2 by C,POLYNOM5:40,FIELD_9:18;
    C3: deg p2 = len p2 - 1 by HURWITZ:def 2;
    len(p1*'p2) <= len p1 + len p2 -' 1 by leng;
    hence (p1*'p2).o = q.o by J,E,C1,C3,C2,XXREAL_0:2,ALGSEQ_1:8;
    end;
  end;
  hence thesis by A;
  end;
then X-c * ((X-a) * (X-b)) = X^3-2 by D,POLYNOM3:def 10;
hence thesis by GROUP_1:def 3;
end;
