Generalized Galois Theory and its Application to Complexity

Abstract We generalize those aspects of classical Galois theory that have to do with the discussion of solvability of problems (namely polynomial equations) relative to auxiliary procedures (e.g. radicals). The underlying structures need no longer be fields, and the problems and procedures more typically arise as algorithmic (e.g. combinatorial) problems. Some of the classical notions and results, e.g. resolvents and discriminants have their natural counterparts. We extend the classical theory mainly in the direction of relations between the group of a problem and the structure and complexity of its solution algorithm. The present paper gives a connected and detailed exposition of this theory, improving and considerably expanding our earlier reports [3, 4]. It now represents a tool for the systematic discussion of the solvability of algorithmic problems, their dependence on structural settings, and the relative merits of solution strategies.