
theorem
for F being Field, p being non zero Polynomial of F
holds card(MonicDivisors p) <= 2|^(deg p)
proof
let F be Field, p be non zero Polynomial of F;
per cases;
suppose p is non constant; then
  deg p > 0 by RATFUNC1:def 2; then
  reconsider u = p as non constant Element of the carrier of Polynom-Ring F
    by POLYNOM3:def 10,RING_4:def 4;
  set E = the SplittingField of u;
  u splits_in E by FIELD_8:def 1; then
  consider a being non zero Element of E, q being Ppoly of E such that
  A: u = a * q by FIELD_4:def 5;
  q is Element of the carrier of Polynom-Ring E by POLYNOM3:def 10; then
  MonicDivisors q = MonicDivisors (a * q) by diveq; then
  C: card(MonicDivisors u) <= card(MonicDivisors q) by NAT_1:43,A,divfin1;
  consider n being Nat such that
  D: card(MonicDivisors q) = n & n <= 2|^(deg q) by divfin2;
  reconsider q1 = a * q as Element of the carrier of Polynom-Ring E
     by POLYNOM3:def 10;
  deg q = deg q1 by RING_5:4 .= deg u by A,FIELD_4:20;
  hence thesis by C,D,XXREAL_0:2;
  end;
suppose A: p is constant;
  now let o be object;
    assume o in MonicDivisors p; then
    consider r being monic Element of the carrier of Polynom-Ring F such that
    B: o = r & r divides p;
    reconsider r as monic Polynomial of F;
    D: deg r <= deg p by B,RING_5:13;
    deg p <= 0 by A,RATFUNC1:def 2;
    then r = 1_.(F) by D,RATFUNC1:def 2,lemconst;
    hence o in {1_.F} by B,TARSKI:def 1;
    end; then
  MonicDivisors p c= {1_.F} & card {1_.F} = 1 by CARD_2:42; then
  C: card(MonicDivisors p) <= 1 by NAT_1:43;
  deg p = 0 by A,RATFUNC1:def 2;
  hence thesis by C,NEWTON:4;
  end;
end;
