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 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