
theorem naH:
for F being non 2-characteristic Field
for p being quadratic Element of the carrier of Polynom-Ring F
holds p is reducible iff DC p is square
proof
let F be non 2-characteristic Field;
let p be quadratic Element of the carrier of Polynom-Ring F;
consider a being non zero Element of F, b,c being Element of F such that
H0: p = <%c,b,a%> by qua5;
Z:now assume DC p is square;
  then consider w being Element of F such that
  A: w^2 = DC p by O_RING_1:def 2;
  A1: DC p = b^2 - 4 '*' a * c by H0,defDC;
  set r1 = (-b + w) * (2 '*' a)", r2 = (-b - w) * (2 '*' a)";
  set q1 = X-r1, q2 = X-r2;
  reconsider qq1 = q1, qq2 = a * q2, p1 = p as Element of Polynom-Ring F
     by POLYNOM3:def 10;
  p = a * (q1 *' q2) by H0,A,A1,lemred
   .= q1 *' (a * q2) by RING_4:10
   .= qq1 * qq2 by POLYNOM3:def 10; then
  B: qq1 divides p1 by GCD_1:def 1;
  C: deg q1 = 1 by HURWITZ:27;
  deg p = 2 by defquadr;
  hence p is reducible by C,B,RING_4:40,RING_4:def 3;
  end;
now assume Y: p is reducible;
  p <> 0_.(F) & p is non unital; then
  consider q being Element of the carrier of Polynom-Ring F such that
  A: q divides p & 1 <= deg q & deg q < deg p by Y,RING_4:41;
  reconsider pp = p, qq = q as Polynomial of F;
  consider rr being Polynomial of F such that
  B: pp = qq *' rr by A,RING_4:1;
  reconsider degq = deg q as Element of NAT by A,INT_1:3;
  E: deg p = 2 by defquadr;
  degq < 1 + 1 by A,defquadr; then
  E0: degq >= 1 & degq <= 1 by A,NAT_1:13; then
  E0a: degq = 1 by XXREAL_0:1;
  consider x1,c1 being Element of F such that
  E1: x1 <> 0.F & q = x1 * rpoly(1,c1) by E0,XXREAL_0:1,HURWITZ:28;
  E2: qq <> 0_.(F) by A,HURWITZ:20;
  rr <> 0_.(F) by B; then
  deg qq + deg rr = 2 by B,E,E2,HURWITZ:23; then
  consider x2,c2 being Element of F such that
  E3: x2 <> 0.F & rr = x2 * rpoly(1,c2) by E0a,HURWITZ:28;
  reconsider x = x1 * x2 as non zero Element of F
                                 by E3,E1,VECTSP_2:def 1,STRUCT_0:def 12;
  qq *' rr
     = x2 * (rpoly(1,c2) *' (x1 * rpoly(1,c1))) by E1,E3,RATFUNC1:6
    .= x2 * (x1 * (rpoly(1,c2) *' rpoly(1,c1))) by RATFUNC1:6
    .= (x2 * x1) * (rpoly(1,c2) *' rpoly(1,c1)) by RING_4:11
    .= x * (rpoly(1,c2) *' rpoly(1,c1)) by GROUP_1:def 12
    .= x * <%c1*c2,-(c1+c2),1.F%> by lemred3z
    .= <%x*(c1*c2),x*(-(c1+c2)),x*1.F%> by qua6;
  then DC p
    = (x*(-(c1+c2)))^2 - 4 '*' x * (x*(c1*c2)) by B,defDC
   .= (x*(-c1+-c2))^2 - 4 '*' x * (x*(c1*c2)) by RLVECT_1:31
   .= (x*(-c1)+x*(-c2))^2 - 4 '*' x * (x*(c1*c2)) by VECTSP_1:def 2
   .= ((x*(-c1))^2 + 2 '*' (x*(-c1)) * (x*(-c2)) + (x*(-c2))^2)
         - 4 '*' x * (x*(c1*c2)) by REALALG2:7
   .= ((x*(-c1))^2 + 2 '*' ((x*(-c1)) * (x*(-c2))) + (x*(-c2))^2)
         - 4 '*' x * (x*(c1*c2)) by REALALG2:5
   .= ((x*(-c1))^2 + 2 '*' (x*((-c1) * (x*(-c2)))) + (x*(-c2))^2)
         - 4 '*' x * (x*(c1*c2)) by GROUP_1:def 3
   .= ((x*(-c1))^2 + 2 '*' (x*((x*(-c2)) * (-c1))) + (x*(-c2))^2)
         - 4 '*' x * (x*(c1*c2)) by GROUP_1:def 12
   .= ((x*(-c1))^2 + 2 '*' (x*(x*((-c2) * (-c1)))) + (x*(-c2))^2)
         - 4 '*' x * (x*(c1*c2)) by GROUP_1:def 3
   .= ((x*(-c1))^2 + 2 '*' (x*(x*(c2 * c1))) + (x*(-c2))^2)
         - 4 '*' x * (x*(c1*c2)) by VECTSP_1:10
   .= ((x*(-c1))^2 + 2 '*' (x*(x*(c1 * c2))) + (x*(-c2))^2)
         - 4 '*' x * (x*(c1*c2)) by GROUP_1:def 12
   .= (((x*(-c1))^2 + 2 '*' (x*(x*(c1 * c2)))) +
      -(4 '*' x * (x*(c1*c2)))) + (x*(-c2))^2 by RLVECT_1:def 3
   .= ((x*(-c1))^2 + (2 '*' (x*(x*(c1 * c2))) +
      -(4 '*' x * (x*(c1*c2))))) + (x*(-c2))^2 by RLVECT_1:def 3
   .= ((x*(-c1))^2 + (2 '*' (x*(x*(c1 * c2))) +
      -(4 '*' (x * (x*(c1*c2)))))) + (x*(-c2))^2 by REALALG2:5
   .= ((x*(-c1))^2 + (2 '*' (x*(x*(c1 * c2))) +
      ((-4) '*' (x * (x*(c1*c2))))) + (x*(-c2))^2) by RING_3:63
   .= ((x*(-c1))^2 + (2 + -4) '*' (x*(x*(c1 * c2)))) +
       (x*(-c2))^2 by RING_3:62
   .= ((x*(-c1))^2 + (-2) '*' (x*(x*(c1 * c2)))) + (x*(-c2))^2
   .= ((x*(-c1))^2 + -(2 '*' (x*(x*(c1 * c2))))) + (x*(-c2))^2
      by RING_3:63
   .= ((x*(-c1))^2 + -(2 '*' (x*(x*((-c1) * (-c2)))))) + (x*(-c2))^2
      by VECTSP_1:10
   .= ((x*(-c1))^2 + -(2 '*' (x*(x*((-c2) * (-c1)))))) + (x*(-c2))^2
      by GROUP_1:def 12
   .= ((x*(-c1))^2 - 2 '*' (x*((x*(-c2)) * (-c1))) + (x*(-c2))^2)
      by GROUP_1:def 3
   .= ((x*(-c1))^2 - 2 '*' (x*((-c1) * (x*(-c2)))) + (x*(-c2))^2)
      by GROUP_1:def 12
   .= ((x*(-c1))^2 - 2 '*' ((x*(-c1)) * (x*(-c2))) + (x*(-c2))^2)
      by GROUP_1:def 3
   .= (x*(-c1))^2 - 2 '*' (x*(-c1)) * (x*(-c2)) + (x*(-c2))^2
      by REALALG2:5
   .= (x*(-c1) - x*(-c2))^2 by REALALG2:8;
  hence DC p is square;
  end;
hence thesis by Z;
end;
