- Person
- Emmy Noether
- Lived
- 1882–1935
- Field
- Mathematics
- From
- German
The story
Invariant theory, the field in which Emmy Noether began her career, studies expressions that stay the same when a group of transformations is applied. A simple example is a metre stick: rotate it and the coordinates of its ends change, but its length does not. A central question was the "finite basis problem": can all the invariants of a given situation be built, by adding and multiplying, from a finite list of basic ones?
Noether's doctoral adviser at Erlangen, Paul Gordan, known as the "king of invariant theory", had solved the problem in 1870 for polynomials in two variables by explicit construction. In 1890 David Hilbert proved a similar result for any number of variables, but his famous proof was non-constructive: it showed a finite basis existed without giving practical information about it. Noether's own 1907 thesis followed Gordan's computational style and ended with a list of more than 300 worked-out invariants. Under the influence of Ernst Fischer, Gordan's successor, she then turned to Hilbert's more abstract methods, and between 1913 and 1916 she applied them to problems including the invariants of finite groups.
In a 1915 paper she solved the finite basis problem for a finite group of transformations acting on a finite-dimensional vector space over a field of characteristic zero. She showed that all the invariants can be generated by homogeneous invariants whose degree is at most the number of elements in the group. This limit is called Noether's bound. Her paper gave two proofs, both of which also work for some fields of prime characteristic, provided the characteristic does not divide the factorial of the group's order.
One case escaped her: fields whose characteristic divides the factorial of the group's order but not the order itself. Whether her bound still held there became a long-standing open question known as "Noether's gap". It was finally settled independently by Fleischmann in 2000 and Fogarty in 2001, who both showed the bound remains true.
In 1926 Noether extended Hilbert's finiteness theorem to representations of a finite group over any field, including the difficult case where the field's characteristic divides the group's order. William Haboush later extended her result to all reductive groups. The same paper introduced the Noether normalization lemma, now a basic tool in commutative algebra and algebraic geometry. Hermann Weyl later contrasted the formal computations of her thesis with the grand conceptual thinking of her mature work, and her invariant theory papers show that transition under way.
Why it matters
Noether's bound and normalization lemma remain standard results taught and used in algebra and algebraic geometry.
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 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's 1921 paper gave us Noetherian ringsIn her 1921 paper Idealtheorie in Ringbereichen, Emmy Noether turned the theory of ideals into a powerful general tool, and objects with her chain condition are still called Noetherian.
- 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 Amalie Noether MacTutor History of Mathematics, University of St Andrews
Another act of tikkun olam. This is part of tikkun olam, the Jewish idea of repairing the world, and every one of us can add to it.