A proof using the Krull-Schmidt theorem shows that certain finite and infinite structures in this discipline have bases ["base"-"ease"] that consist of conjugates of single elements; those normal bases always exist for this theory's namesake structures.
This discipline's namesake "descent" technique was used on differential algebraic structures arising from solutions to differential equations to generate a Picard-Vessiot theory. A proof using the Krull-Schmidt theorem shows that certain finite and infinite structures in this discipline have bases ["base"-"ease"] that consist of conjugates of single elements; those normal bases always exist for this theory's namesake structures. For certain finite structures, there exists a bijection between their intermediate fields and (*) subgroups of this theory's namesake groups according to this discipline's fundamental theorem. This theory, which is concerned with automorphisms of field extensions, is used to show that higher-order equations generally lack solvability. For 10 points, identify this theory from abstract algebra named for a Frenchman who died in a duel at the age of 20.
This discipline's namesake "descent" technique was used on differential algebraic structures arising from solutions to differential equations to generate a Picard-Vessiot theory.
Galois theory