
theorem bas3R:
{ 1, 3-Root(2), 3-Root(2)^2 }
is Basis of VecSp(FAdj(F_Rat,{3-Root(2)}),F_Rat)
proof
set F = F_Rat, M = { 1.F_Real, 3-Root(2), 3-Root(2)^2 };
B: now let o be object;
   assume o in Base 3-Root(2); then
   consider n being Element of NAT such that
   B1: o = 3-Root(2)|^n & n < deg MinPoly(3-Root(2),F);
   B2: n < 2 + 1 by LL,B1,LL2,FIELD_6:52;
   n <= 2 implies n = 0 or ... or n = 2; then
   per cases by B2,NAT_1:13;
   suppose n = 0;
    then o = 1_F_Real by B1,BINOM:8;
    hence o in M by ENUMSET1:def 1;
    end;
   suppose n = 1;
    then o = 3-Root(2) by B1,BINOM:8;
    hence o in M by ENUMSET1:def 1;
    end;
   suppose n = 2;
    then o = 3-Root(2)|^(1+1) by B1
          .= 3-Root(2)|^1 * 3-Root(2)|^1 by BINOM:10
          .= 3-Root(2)|^1 * 3-Root(2) by BINOM:8
          .= 3-Root(2) * 3-Root(2) by BINOM:8
          .= 3-Root(2)^2 by O_RING_1:def 1;
    hence o in M by ENUMSET1:def 1;
    end;
   end;
now let o be object;
  assume o in M; then
  per cases by ENUMSET1:def 1;
  suppose o = 1.F_Real;
    then o = 1_F_Real .= 3-Root(2)|^0 by BINOM:8;
    hence o in Base 3-Root(2) by LL,minpolzeta;
    end;
  suppose o = 3-Root(2);
    then o = 3-Root(2)|^1 by BINOM:8;
    hence o in Base 3-Root(2) by LL,minpolzeta;
    end;
  suppose o = 3-Root(2)^2;
    then o = 3-Root(2) * 3-Root(2) by O_RING_1:def 1
          .= 3-Root(2)|^1 * 3-Root(2) by BINOM:8
          .= 3-Root(2)|^1 * 3-Root(2)|^1 by BINOM:8
          .= 3-Root(2)|^(1+1) by BINOM:10;
    hence o in Base 3-Root(2) by LL,minpolzeta;
    end;
  end;
then Base 3-Root(2) = M by B,TARSKI:2;
hence thesis by FIELD_6:65;
end;
