Atlas of Fundamental Concepts

A polynomial over a field may require adjoining roots. Algebraic, separable, and normal extensions isolate the conditions under which all embeddings and roots behave symmetrically; their conjunction is a Galois extension. Base-fixing field automorphisms form the Galois group. The fundamental theorem reverses inclusion between intermediate fields and subgroups, converting field questions into group structure. Solvable groups then characterize solvability by radicals under the standard hypotheses.