
theorem
for F being Field,
    E being FieldExtension of F
for p,q being Element of the carrier of Polynom-Ring F
for p1,q1 being Element of the carrier of Polynom-Ring E
st p1 = p & q1 = q holds q1 divides p1 iff q divides p
proof
let F be Field, E be FieldExtension of F;
let p be Element of the carrier of Polynom-Ring F;
let q be Element of the carrier of Polynom-Ring F;
let p1,q1 be Element of the carrier of Polynom-Ring E;
assume AS: p1 = p & q1 = q;
per cases;
suppose q is zero; then
   S1: q = 0_.(F) by UPROOTS:def 5; then
   S2: q1 = 0_.(E) by AS,FIELD_4:12;
A: now assume q divides p; then
   consider r being Polynomial of F such that
   A1: (0_.(F)) *' r = p by S1,RING_4:1;
   q1 *' p1 = p1 by S2,AS,A1,FIELD_4:12;
   hence q1 divides p1 by RING_4:1;
   end;
   now assume q1 divides p1;then
   consider r being Polynomial of E such that
   A1: (0_.(E)) *' r = p1 by S2,RING_4:1;
   q *' p = p by S1,AS,A1,FIELD_4:12;
   hence q divides p by RING_4:1;
   end;
   hence thesis by A;
   end;
suppose S: q is non zero; then
   q <> 0_.(F); then
   q <> 0_.(E) by FIELD_4:12; then
   S1: q1 is non zero by AS,UPROOTS:def 5;
A: now assume A1: q1 divides p1;
   p gcd q = p1 gcd q1 by AS,lemgcd
          .= NormPolynomial q1 by S1,A1,lemgcdn
          .= NormPolynomial q by AS,FIELD_6:24;
   hence q divides p by RING_4:25,RING_4:52;
   end;
now assume A1: q divides p;
   p1 gcd q1 = p gcd q by AS,lemgcd
            .= NormPolynomial q by S,A1,lemgcdn
            .= NormPolynomial q1 by AS,FIELD_6:24;
   hence q1 divides p1 by RING_4:25,RING_4:52;
   end;
hence thesis by A;
end;
end;
