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.
In 1932 Emil Artin and I received the Ackermann–Teubner Memorial Award, and that September I gave a plenary address at the International Congress of Mathematicians in Zürich — recognition of my research, long overdue.
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.
At Göttingen I supervised more than a dozen doctoral students and led a wider circle — Krull, Köthe, van der Waerden — whose work and textbooks spread my approach to abstract algebra across the field.
I taught without a lesson plan: my Göttingen lectures were live research sessions where I worked through open problems with students. Several of my key results came out of them, and students' notes became textbooks.
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.
Work on hypercomplex numbers and group representations had stayed disparate for decades. I united it into the first general representation theory of groups and algebras — one arithmetic theory of modules and ideals.
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.
I made the ascending and descending chain conditions central tools: one simple finiteness assumption on ideals gives finite generation, primary decomposition and the structure of Dedekind domains.
