
theorem
for F being Field
for p being non constant Element of the carrier of Polynom-Ring F
for a being non zero Element of F
holds a * p is separable iff p is separable
proof
let F be Field, p be non constant Element of the carrier of Polynom-Ring F;
let c be non zero Element of F;
A: now assume AS: p is separable;
   now let E be FieldExtension of F;
     F is Subfield of E & c <> 0.F by FIELD_4:7; then
     H: c <> 0.E & @(c,E) = c by EC_PF_1:def 1,FIELD_7:def 4; then
     K: @(c,E) is non zero;
     reconsider pE = p as Polynomial of E by FIELD_4:8;
     I: @(c,E) * pE = c * p by H,ll;
     assume (c*p) splits_in E; then
     consider b being non zero Element of E, r being Ppoly of E such that
     J: c * p = b * r by FIELD_4:def 5;
     @(c,E) * pE splits_in E by I,J,FIELD_4:def 5; then
     consider b being non zero Element of E, r being Ppoly of E such that
     J: pE = b * r by K,FIELD_8:9,FIELD_4:def 5;
     A: p splits_in E by J,FIELD_4:def 5;
     thus for a being Element of E
          st a is_a_root_of (c*p),E holds multiplicity(c*p,a) = 1
       proof
       let a be Element of E;
       assume a is_a_root_of (c*p),E;
       then 0.E = Ext_eval(c*p,a) by FIELD_4:def 2
               .= @(c,E) * Ext_eval(p,a) by FIELD_7:def 4,REALALG3:16;
       then Ext_eval(p,a) = 0.E by H,VECTSP_2:def 1;
       then multiplicity(p,a) = 1 by AS,A,ThSep0,FIELD_4:def 2;
       hence multiplicity(c*p,a) = 1 by lems1;
       end;
     end;
   hence c*p is separable by ThSep0;
   end;
now assume AS: c * p is separable;
   now let E be FieldExtension of F;
     F is Subfield of E & c <> 0.F by FIELD_4:7; then
     H: c <> 0.E &  @(c,E) = c by EC_PF_1:def 1,FIELD_7:def 4; then
     K: @(c,E) is non zero;
     assume p splits_in E; then
     consider b being non zero Element of E, r being Ppoly of E such that
     J: p = b * r by FIELD_4:def 5;
     c * p = @(c,E) * (b * r) by H,J,ll
          .= (@(c,E) * b) * r by RING_4:11; then
     A: (c*p) splits_in E by K,FIELD_4:def 5;
     thus for a being Element of E
          st a is_a_root_of p,E holds multiplicity(p,a) = 1
       proof
       let a be Element of E;
       assume B: a is_a_root_of p,E;
       Ext_eval(c*p,a)
            = @(c,E) * Ext_eval(p,a) by REALALG3:16,FIELD_7:def 4
           .= @(c,E) * 0.E by B,FIELD_4:def 2;
       then multiplicity(c*p,a) = 1 by AS,A,ThSep0,FIELD_4:def 2;
       hence multiplicity(p,a) = 1 by lems1;
       end;
     end;
  hence p is separable by ThSep0;
  end;
hence thesis by A;
end;
