Polynomial Computability of Fields of Algebraic Numbers

Abstract

We prove that the field of complex algebraic numbers and the ordered field of real algebraic numbers have isomorphic presentations computable in polynomial time. For these presentations, new algorithms are found for evaluation of polynomials and solving equations of one unknown. It is proved that all best known presentations for these fields produce polynomially computable structures or quotient-structures such that there exists an isomorphism between them polynomially computable in both directions.