 reserve n for Nat;
 reserve F for Field,
         p for irreducible Element of the carrier of Polynom-Ring F,
         f for Element of the carrier of Polynom-Ring F,
         a for Element of F;

theorem Th45:
   emb p is isomorphism implies p is with_roots
   proof
     set h = emb p;
     assume A1: emb p is isomorphism; then
     reconsider K = KroneckerField(F,p) as F-isomorphic F-homomorphic Ring
       by RING_3:def 4;
     reconsider h as Isomorphism of F,K by A1;
A2:   Roots (PolyHom h).p = {h.a where a is Element of F : a in Roots p}
       by Th39;
     KrRoot p is_a_root_of emb(p,p) by Th43; then
     KrRoot p in Roots (PolyHom h).p by POLYNOM5:def 10; then
     ex a being Element of F st (KrRoot p) = h.a & a in Roots p by A2;
     hence thesis;
   end;
