- Person
- Emmy Noether
- Lived
- 1882–1935
- Field
- Mathematics
- From
- German
The story
During the nineteenth and early twentieth centuries, mathematicians produced many results about hypercomplex numbers and about representations of groups, ways of describing abstract symmetries by matrices and linear transformations. The results were valuable but disparate, proved by different methods for different cases.
Emmy Noether's colleague Hermann Weyl divided her work into three periods. The third, from 1927 to 1935, was devoted to noncommutative algebras, their representations by linear transformations and their applications to number theory. She developed these ideas largely in her Göttingen lectures, including courses on group theory and hypercomplex numbers in the winter of 1924–25 and on hypercomplex quantities and representation theory in the winter of 1927–28. Some of her most important results emerged in these lectures, and her students' notes became the basis of textbooks.
Noether united the earlier results and gave the first general representation theory of groups and algebras. Her key move was to bring everything into the framework she had already built for ideal theory: she subsumed the structure theory of associative algebras and the representation theory of groups into a single arithmetic theory of modules and ideals in rings satisfying ascending chain conditions. Instead of studying each representation separately, mathematicians could now treat representations as modules and apply general algebraic tools to them. This single work was said to have ushered in a new period in modern algebra and to have been of fundamental importance for its development.
She carried the approach with her into exile. In a letter to Helmut Hasse in March 1934, MacTutor records, she wrote that she had begun her lectures at the Institute for Advanced Study in Princeton with representation modules and groups with operators, giving Princeton its first thorough algebraic treatment of the subject. Others in her circle, such as Gottfried Köthe, extended the theory of hypercomplex quantities using her methods and those of Wolfgang Krull.
The module-theoretic viewpoint she introduced is now the standard way representation theory is taught. Representation theory is the mathematical language of symmetry, the same idea that lies behind her famous theorem in physics, and her unified treatment helped make it a shared tool for algebraists, number theorists and physicists alike.
Why it matters
Noether's unified representation theory gave mathematicians and physicists a common language for symmetry that is used throughout modern science.
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'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 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
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