Explicit Computation of Isomorphisms between Finite Fields

Although it is easy to prove that two finite fields having the same cardinality are isomorphic, the proof uses embeddings into an algebraic closure (or at least into a common overfield), hence is not constructive, and so does not provide explicit isomorphisms. We give algorithms to solve this problem efficiently in practice, and as an application, we also give an algorithm for factoring a polynomial P P[X] over a finite extension of P