
theorem pirred:
for p being Prime
for F being p-characteristic Field
for a being Element of F
st not ex b being Element of F st b|^p = a holds X^(p,a) is irreducible
proof
let p be Prime, F be p-characteristic Field, aF be Element of F;
assume AS: not ex b being Element of F st b|^p = aF;
consider E being FieldExtension of F such that
A0: X^(p,aF) is_with_roots_in E by FIELD_5:30;
    consider bE being Element of E such that
A1: bE is_a_root_of X^(p,aF),E by A0,FIELD_4:def 3;
    F is Subfield of E by FIELD_4:7; then
    the carrier of F c= the carrier of E by EC_PF_1:def 1; then
    reconsider aE = aF as Element of E;
E: F is Subring of E & p is Element of NAT by ORDINAL1:def 12,FIELD_4:def 1;
H: X^(p,aF) = X^(p,aE) by split2;
   0.E = Ext_eval(X^(p,aF),bE) by A1,FIELD_4:def 2
      .= eval(X^(p,aE),bE) by split2,FIELD_4:26; then
   bE is_a_root_of X^(p,aE); then
B: bE|^p = aE by split1; then
U: bE|^p = aE & X^(p,aE) = (X-bE)|^p by split0;
C: now assume bE in the carrier of F; then
   reconsider bF = bE as Element of F;
   bF|^p = aF by B,E,FIELD_6:19;
   hence contradiction by AS;
   end;
   (0.F)|^p = 0.F; then
D: aF <> 0.F by AS;
now assume X^(p,aF) is reducible; then
  consider qF being monic Element of the carrier of Polynom-Ring F such that
  C1: qF divides X^(p,aF) & 1 <= deg qF & deg qF < deg X^(p,aF) by R441;
  the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E by FIELD_4:10;
  then reconsider qE = qF as Element of the carrier of Polynom-Ring E;
  LC qE = LC qF by FIELD_8:5
       .= 1.F by RATFUNC1:def 7 .= 1.E by E,C0SP1:def 3; then
  C2: qE is monic by RATFUNC1:def 7;
  qE divides (X-bE)`^p by U,H,C1,FIELD_14:48; then
  consider l being Nat such that C3: l <= p & qE = (X-bE)`^l by C2,ro1;
  C4: l <> 0 & l < p
      proof
      now assume l = 0;
        then qE = 1_.(E) by C3,POLYNOM5:15 .= 1_.(F) by FIELD_4:14;
        hence contradiction by C1,RATFUNC1:def 2;
        end;
      hence l <> 0;
      thus thesis by U,H,C3,C1,XXREAL_0:1;
      end;
  C5: bE|^l in the carrier of F
      proof
      D: qF.0 = ((X-bE).0)|^l by C3,C4,t3
             .= (-power(E).(bE,0+1))|^l by HURWITZ:25
             .= (-(power(E).(bE,0) * bE))|^l by GROUP_1:def 7
             .= (-(1_E * bE))|^l by GROUP_1:def 7
             .= (-bE)|^l;
      per cases;
      suppose l is even;
        then qF.0 = bE|^l by D,teven;
        hence thesis;
        end;
      suppose l is odd;
        then reconsider c = -(bE|^l) as Element of F by D,todd;
        -c = --(bE|^l) by E,FIELD_6:17;
        hence thesis;
        end;
      end;
  l gcd p = 1 by C4,gcdp; then
  consider s,t being Integer such that C6: 1 = s * p + t * l by NAT_D:68;
  set M1 = MultGroup E;
  reconsider M2 = MultGroup F as Subgroup of M1 by MGsub;
  H1: the carrier of E = (the carrier of M1) \/ {0.E} &
      the carrier of F = (the carrier of M2) \/ {0.F} by UNIROOTS:15;
  bE <> 0.F by C; then
  bE <> 0.E & aE <> 0.E by D,E,C0SP1:def 3; then
  not bE in {0.E} & not aE in {0.E} by TARSKI:def 1; then
  reconsider b = bE, a = aE as Element of M1 by H1,XBOOLE_0:def 3;
  H2: bE|^l = b|^l by MG;
  bE <> 0.F by C; then
  bE is non zero by E,C0SP1:def 3; then
  bE|^l <> 0.F by E,C0SP1:def 3; then
  not b|^l in {0.F} & not a in {0.F} by H2,D,TARSKI:def 1; then
  reconsider x = b|^l, y = a as Element of M2 by H2,C5,H1,XBOOLE_0:def 3;
  C7: b|^p = a by B,MG;
  C8: a|^s = y|^s & x|^t = (b|^l)|^t by GROUP_4:2;
  b = b|^(s * p + t * l) by C6,GROUP_1:26
   .= b|^(s * p) * b|^(t * l) by GROUP_1:33
   .= (b|^p) |^ s * b|^(t * l) by GROUP_1:35
   .= a|^s * (b|^l)|^t by C7,GROUP_1:35
   .= y|^s * x|^t by C8,GROUP_2:43;
  hence contradiction by C,H1,XBOOLE_0:def 3;
  end;
hence thesis;
end;
