
theorem PrimAUT:
for F being prime Field
for f being Automorphism of F holds f = id F
proof
let F be prime Field, f be Automorphism of F;
A: F = PrimeField F by RING_3:96;
per cases by RING_3:86;
suppose Char F = 0; then
  F, F_Rat are_isomorphic by A,RING_3:113; then
  consider h being Function of F, F_Rat such that
  B: h is isomorphism;
  reconsider FRat = F_Rat as F-isomorphic Field by B,RING_3:def 4;
  reconsider h as Isomorphism of F,FRat by B;
  reconsider g2 = h" as Isomorphism of FRat,F by RING_3:73;
  g2 is isomorphism; then
  reconsider F as FRat-isomorphic Field by RING_3:def 4;
  D: rng h = the carrier of FRat by FUNCT_2:def 3;
  reconsider g1 = f * g2 as Function of FRat,F;
  reconsider g = h * (f * g2) as Function of FRat,FRat;
  E: g1 is isomorphism;
  h" * g = (h" * h) * (f * h") by T2
        .= (id F) * (f * h") by D,FUNCT_2:29
        .= f * h"; then
  (h" * g) * h = f * (h" * h) by T2
              .= f * (id F) by D,FUNCT_2:29
              .= f; then
  f = (h" * id(FRat)) * h by E,RING_3:100
   .= id F by D,FUNCT_2:29;
  hence thesis;
  end;
suppose Char F is prime; then
  reconsider p = Char F as Prime;
  F, Z/p are_isomorphic by A,RING_3:114; then
  consider h being Function of F,Z/p such that
  B: h is isomorphism;
  reconsider Zp = Z/p as F-isomorphic Field by B,RING_3:def 4;
  reconsider h as Isomorphism of F,Zp by B;
  reconsider g2 = h" as Isomorphism of Zp,F by RING_3:73;
  g2 is isomorphism; then
  reconsider F as Zp-isomorphic Field by RING_3:def 4;
  D: rng h = the carrier of Zp by FUNCT_2:def 3;
  reconsider g1 = f * g2 as Function of Zp,F;
  reconsider g = h * (f * g2) as Function of Zp,Zp;
  E: g1 is isomorphism;
  h" * g = (h" * h) * (f * h") by T2
        .= (id F) * (f * h") by D,FUNCT_2:29
        .= f * h"; then
  (h" * g) * h = f * (h" * h) by T2
              .= f * (id F) by D,FUNCT_2:29
              .= f; then
  f = (h" * id(Z/p)) * h by E,RING_3:107
   .= id F by D,FUNCT_2:29;
  hence thesis;
  end;
end;
