
theorem mulzero:
for F being Field,
    E being FieldExtension of F
for p being non zero Element of the carrier of Polynom-Ring F
for a being Element of E
holds a is_a_root_of p,E iff multiplicity(p,a) >= 1
proof
let F be Field, E be FieldExtension of F;
let p be non zero Element of the carrier of Polynom-Ring F;
let a be Element of E;
the carrier of Polynom-Ring F c= the carrier of Polynom-Ring E
  by FIELD_4:10; then
reconsider q = p as Element of the carrier of Polynom-Ring E;
p <> 0_.(F); then
q <> 0_.(E) by FIELD_4:12; then
reconsider q as non zero Element of the carrier of Polynom-Ring E
   by UPROOTS:def 5;
A: now assume multiplicity(p,a) >= 1; then
   B: multiplicity(q,a) >= 1 by FIELD_14:def 6;
   (X-a)`^multiplicity(q,a) divides q by FIELD_14:67; then
   (X-a)`^1 divides q by B,FIELD_14:40; then
   (X-a) divides q by POLYNOM5:16; then
   0.E = eval(q,a) by RING_5:11 .= Ext_eval(p,a) by FIELD_4:26;
   hence a is_a_root_of p,E by FIELD_4:def 2;
   end;
now assume a is_a_root_of p,E; then
   0.E = Ext_eval(p,a) by FIELD_4:def 2 .= eval(q,a) by FIELD_4:26; then
   (X-a) divides q by RING_5:11; then
   (X-a)`^1 divides q by POLYNOM5:16; then
   multiplicity(q,a) >= 1 by mulzero1;
   hence multiplicity(p,a) >= 1 by FIELD_14:def 6;
   end;
hence thesis by A;
end;
