
theorem divfin1:
for F being Field,
    E being FieldExtension of F
for p being Polynomial of F for q being Polynomial of E
st q = p holds MonicDivisors p c= MonicDivisors q
proof
let F be Field, E be FieldExtension of F;
let p be Polynomial of F; let q be Polynomial of E;
assume AS: q = p;
now let o be object;
  assume o in MonicDivisors p; then
  consider rp being monic Element of the carrier of Polynom-Ring F such that
  A: o = rp & rp divides p;
  consider up being Polynomial of F such that
  B: p = rp *' up by A,RING_4:1;
  reconsider rq = rp as Polynomial of E by FIELD_4:8;
  C: F is Subfield of E by FIELD_4:7;
  LC rq = LC rp by FIELD_8:5 .= 1.F by RATFUNC1:def 7
       .= 1.E by C,EC_PF_1:def 1; then
  reconsider rq as monic Element of the carrier of Polynom-Ring E
     by RATFUNC1:def 7,POLYNOM3:def 10;
  reconsider uq = up as Polynomial of E by FIELD_4:8;
  p = rq *' uq by B,FIELD_4:17;
  then rq divides q by AS,RING_4:1;
  hence o in MonicDivisors q by A;
  end;
hence thesis;
end;
