How can a complex square root be computed in an optimal way?

Abstract

We give an optimality analysis for computations of complex square roots in real arithmetic by certain computation tress that use real square root operations. Improving standard elementary geometric constructions Schönhage suggests better methods which will be shown to be unimprobable. The iteration of such a procedure for 2^k-th roots is however “improvable,” and an improved version of it can also be shown to be “unimprovable.” In particular, repeated usage of an optimal square root procedure does not yield an optimal one for 2^k-th roots.

To answer this kind of questions about resolution by real radicals we apply methods of real algebra which lead into the theory of real field, ring, and integral ring extensions.