- Person
- Emmy Noether
- Lived
- 1882–1935
- Field
- Mathematics
- From
- German
The story
After the First World War Emmy Noether, by then lecturing at Göttingen, turned from invariant theory to the foundations of algebra. Her habilitation was finally approved in 1919, and in 1920, with her protégé Werner Schmeidler, she published a paper on the theory of ideals that defined left and right ideals in a ring. Hermann Weyl later said that in this period she began work that changed the face of algebra.
Her classic paper Idealtheorie in Ringbereichen (Theory of Ideals in Ring Domains) appeared in 1921. Before it, most results in commutative algebra were restricted to particular examples, such as polynomial rings over fields or rings of algebraic integers. Noether's paper gave one of the first general definitions of a commutative ring and is regarded as the foundation of general commutative ring theory. Instead of calculating with particular equations, she looked for the general structures that make such calculations unnecessary.
Her key tool was the ascending chain condition: any increasing chain of ideals must eventually stop growing. From this simple finiteness condition she drew powerful results. She proved that in such a ring every ideal is finitely generated, and she proved in full generality the Lasker–Noether theorem, which says that ideals can be decomposed into primary ideals. It extended a result that Emanuel Lasker, the mathematician and world chess champion, had proved for polynomial rings, and it can be seen as a far-reaching generalisation of the fact that every whole number factors uniquely into primes. The algebraist Irving Kaplansky called the work revolutionary.
A few years later she characterised, by five simple conditions, the rings in which ideals factor uniquely into prime ideals, now called Dedekind domains. The same paper contains what are now known as the isomorphism theorems, among the most basic results of abstract algebra. Her student Bartel van der Waerden spread these ideas worldwide in his 1931 textbook Moderne Algebra.
Objects that satisfy her ascending chain condition are named Noetherian in her honour, and in 1943 the French mathematician Claude Chevalley coined the term Noetherian ring. Today there are Noetherian rings, modules, groups and spaces, and the historian Jeremy Gray has observed that every textbook on abstract algebra shows her influence, because mathematicians simply do ring theory her way.
Why it matters
Noether's ideal theory became a foundation of modern algebra and algebraic geometry, used by mathematicians everywhere.
About Emmy Noether
Erlangen-born mathematician who linked symmetry to conservation laws and reshaped abstract algebra, despite years of unpaid work as a woman and dismissal by the Nazis as a Jew.
More about Emmy Noether
- Emmy Noether's bound solved a problem in invariant theoryIn 1915 Emmy Noether solved the finite basis problem for finite groups, proving a degree limit now called Noether's bound. A last open case was settled only in 2000 and 2001.
- Emmy Noether helped classify division algebrasWith Richard Brauer and Helmut Hasse, Emmy Noether proved the Brauer–Hasse–Noether theorem, which allows all central division algebras over a number field to be classified.
- Emmy Noether helped create algebraic topologyAttending lectures by Pavel Alexandrov and Heinz Hopf in 1926–27, Emmy Noether suggested the idea of homology groups, helping turn combinatorial topology into algebraic topology.
- Emmy Noether built a school of brilliant studentsEmmy Noether's Göttingen circle, the Noether boys, included doctoral students such as Grete Hermann, Ernst Witt, Max Deuring and Jacob Levitzki, who later taught in Jerusalem.
- Emmy Noether linked symmetry to the laws of conservationIn 1918 Emmy Noether proved that every continuous symmetry in physics comes with a conserved quantity, tying conservation of energy and momentum to symmetry.
- Emmy Noether unified group representation theoryEmmy Noether gave the first general representation theory of groups and algebras, combining scattered earlier results into a single theory of modules and ideals.
- Emmy Noether worked seven years without payOne of only two women among 986 students at Erlangen, Emmy Noether earned her doctorate in 1907 and then worked at the university's Mathematical Institute for seven years without pay.
- Dismissed by the Nazis, Emmy Noether taught at homeAfter the Nazis barred her from teaching in 1933, Emmy Noether gathered students in her apartment to keep doing mathematics, unruffled even when one arrived in SA uniform.
Sources
- Emmy Noether Wikipedia
- Emmy Noether MacTutor History of Mathematics, University of St Andrews
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