In 1918 I reversed the usual Galois question — given a field and a group, is there an extension with that Galois group? — reduced it to Noether's problem, and proved it for n = 2, 3 and 4.
I solved the finite basis problem for finite groups in characteristic zero (1915), bounding generator degrees by the group's order, and in 1926 extended Hilbert's theorem and introduced the normalization lemma.
Attending Alexandrov's and Hopf's lectures in 1926–1927, I suggested that topology be studied algebraically through homology groups — the idea that helped turn combinatorial topology into algebraic topology.
With Emil Artin, Richard Brauer and Helmut Hasse I founded the theory of central simple algebras; our local-global theorem made it possible to classify the division algebras over a number field.
My theorem lets science researchers determine conserved quantities from the observed symmetries of physical laws, and test a new theory by checking its predicted conservation in experiments.
