Computing Roots of Unity in Fields

The paper is in four sections. First we provide some necessary background to computing in fields; then follows a collection of algebraic lemmas. In Section 3 we prove the theorem and in Section 4 append useful information on related decision problems. These theorems are straightforward contributions to Computable Algebra, to work on fields and (non finitely presented) groups (see Sections 1 and 4 for references). Such arbitrary degree results are desirable for Recursion Theory since they establish that the Turing degree, its natural classification of complexity, is relevant to algorithmic questions as they arise in Algebra. And since our fields and groups are systems within the complex numbers C these particular insoluble decision problems are of a particularly elementary nature. We are happy to acknowledge the support of the Matematisk institutt, Universitetet i Oslo, and several convivial conversations with friends in its Algebra group; also the patronage of our present institutions in Uppsala and Amsterdam respectively.