
theorem naH2:
for F being non 2-characteristic Field
for a being Element of F holds X^2-a is reducible iff a is square
proof
let F be non 2-characteristic Field; let a be Element of F;
H: 2 '*' 1.F <> 0.F & 4 '*' 1.F <> 0.F by ch2,ch4;
B: now assume X^2-a is reducible;
   then DC <%-a,0.F,1.F%> is square by naH;
   then - 4 '*' (-a) is square by defDCpq;
   then 4 '*' (--a) is square by REALALG2:6; then
   consider w being Element of F such that
C: w^2 = 4 '*' a by O_RING_1:def 2;
   w * w = 4 '*' (1.F * a) by C,O_RING_1:def 1
        .= (4 '*' 1.F) * a by REALALG2:5
        .= a * (4 '*' 1.F) by GROUP_1:def 12; then
   (w * w) * (4 '*' 1.F)"
         = a * ((4 '*' 1.F) * (4 '*' 1.F)") by GROUP_1:def 3
        .= a * ((4 '*' 1.F)" * (4 '*' 1.F)) by GROUP_1:def 12
        .= a * 1.F by H,VECTSP_1:def 10; then
   a = (w * w) * ((2*2) '*' 1.F)"
    .= (w * w) * ((2 '*' 1.F) * (2 '*' 1.F))" by RING_3:67
    .= (w * w) * ((2 '*' 1.F)" * (2 '*' 1.F)") by H,VECTSP_2:11
    .= w * (w * ((2 '*' 1.F)" * (2 '*' 1.F)")) by GROUP_1:def 3
    .= w * ((w * (2 '*' 1.F)") * (2 '*' 1.F)") by GROUP_1:def 3
    .= w * ((2 '*' 1.F)" * (w * (2 '*' 1.F)")) by GROUP_1:def 12
    .= (w * (2 '*' 1.F)") * (w * (2 '*' 1.F)") by GROUP_1:def 3
    .= (w * (2 '*' 1.F)")^2 by O_RING_1:def 1;
   hence a is square;
   end;
now assume a is square; then
   consider w being Element of F such that
C: w^2 = a by O_RING_1:def 2;
   (2 '*' w)^2 = (2 '*' w) * (2 '*' w) by O_RING_1:def 1
              .= 2 '*' (w * (2 '*' w)) by REALALG2:5
              .= 2 '*' ((2 '*' w) * w) by GROUP_1:def 12
              .= 2 '*' (2 '*' (w * w)) by REALALG2:5
              .= (2*2) '*' (w * w) by RING_3:65
              .= 4 '*' a by C,O_RING_1:def 1;
   then 4 '*' (--a) is square;
   then - 4 '*' (-a) is square by REALALG2:6;
   then DC <%-a,0.F,1.F%> is square  by defDCpq;
   hence X^2-a is reducible by naH;
   end;
hence thesis by B;
end;
