
theorem lemBas01:
for F being Field,
    E being FieldExtension of F
for a being F-algebraic Element of E holds
not a in F iff
for p being non zero Polynomial of F st Ext_eval(p,a) = 0.E holds deg p >= 2
proof
let F be Field, E be FieldExtension of F, a be F-algebraic Element of E;
H2: F is Subfield of E by FIELD_4:7; then
H3: the carrier of F c= the carrier of E & 1.E = 1.F & 0.E = 0.F
    by EC_PF_1:def 1;
Z: now assume AS: not a in F;
   now let p be non zero Polynomial of F;
     assume A: Ext_eval(p,a) = 0.E;
     reconsider q = p as Element of the carrier of Polynom-Ring F
           by POLYNOM3:def 10;
     now assume deg p < 2; then
       per cases by NAT_1:23;
       suppose deg p = 1; then
         consider x,z being Element of F such that
         A1: x <> 0.F & p = x * rpoly(1,z) by HURWITZ:28;
         reconsider xE = x, zE = z as Element of E by H3;
         A2: p is Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
         the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E
           by FIELD_4:10;
         then reconsider pE = p as Element of the carrier of Polynom-Ring E
            by A2;
         A4: x|F = xE|E & rpoly(1,zE) = rpoly(1,z) by FIELD_4:21,FIELD_6:23;
         A3: xE * rpoly(1,zE) = (xE|E) *' rpoly(1,zE) by FIELD_8:2
                             .= (x|F) *' rpoly(1,z) by A4,FIELD_4:17
                             .= x * rpoly(1,z) by FIELD_8:2;
         A4: xE <> 0.E by A1,H2,EC_PF_1:def 1;
         0.E = eval(pE,a) by A,A2,FIELD_4:26
            .= xE * eval(rpoly(1,zE),a) by A1,A3,POLYNOM5:30;
         then 0.E = eval(rpoly(1,zE),a) by A4,VECTSP_2:def 1
                 .= a - zE by HURWITZ:29;
         then a = z by RLVECT_1:21;
         hence contradiction by AS;
         end;
       suppose deg p = 0; then
         consider b being Element of F such that
         A1: q = b|F by RING_4:def 4,RING_4:20;
         A2: b <> 0.F by A1;
         reconsider bE = b as Element of E by H3;
         A3: bE|E = q &
             bE|E is Element of the carrier of Polynom-Ring E
             by A1,FIELD_6:23,POLYNOM3:def 10;
         A4: deg(bE|E) = 0 by A2,H3,RING_4:21;
         0.E = eval(bE|E,a) by A,A3,FIELD_4:26;
         hence contradiction by A4,RATFUNC1:8;
         end;
       end;
     hence deg p >= 2;
     end;
   hence for p being non zero Polynomial of F
                         st Ext_eval(p,a) = 0.E holds deg p >= 2;
   end;
now assume a in F;
  then reconsider b = a as Element of F;
  thus ex p being non zero Polynomial of F st Ext_eval(p,a) = 0.E & deg p < 2
    proof
    take p = X-b;
    I: rpoly(1,a) is Element of the carrier of Polynom-Ring E &
       rpoly(1,b) is Element of the carrier of Polynom-Ring F
       by POLYNOM3:def 10;
    thus Ext_eval(p,a)
       = Ext_eval(rpoly(1,b),a) by FIELD_9:def 2
      .= eval(rpoly(1,a),a) by FIELD_4:21,I,FIELD_4:26
      .= a - a by HURWITZ:29
      .= 0.E by RLVECT_1:5;
    deg p = deg rpoly(1,b) by FIELD_9:def 2 .= 1 by HURWITZ:27;
    hence deg p < 2;
    end;
  end;
hence thesis by Z;
end;
