Computational Complexity of Algebraic Functions

We consider algebraic functions that are rational functions of roots (of various degrees) of rational functions of indeterminates. We associate a cost C(d) with the extraction of a dth root and assume that C satisfies certain natural axioms. We show that the minimum cost of computing a finite set of algebraic functions of the form considered is C(d1) + ... + C(dr), where d1...dr are the torsion orders of the Galois group of the extension generated by the functions. © 1981.