- Person
- Emmy Noether
- Lived
- 1882–1935
- Field
- Mathematics
- From
- German
The story
By the late 1920s Emmy Noether had transformed commutative algebra with her theory of ideals. Hermann Weyl described the third period of her work, from 1927 until her death in 1935, as devoted to noncommutative algebras, their representations and their application to the arithmetic of number fields. Beginning in 1927 she worked closely with Emil Artin, Richard Brauer and Helmut Hasse, and together they founded the theory of central simple algebras.
A key object in this theory is a division algebra, an algebraic system in which, as with ordinary numbers, division is always possible, but in which multiplication need not be commutative. Number theorists wanted to know which such algebras can exist over a given number field. Noether, Brauer and Hasse answered the question in a joint paper, "Beweis eines Hauptsatzes in der Theorie der Algebren" (Proof of a Main Theorem in the Theory of Algebras), published in 1932. MacTutor describes it as a beautiful paper.
They proved two important theorems. The first is a local-global principle: if a finite-dimensional central division algebra over a number field splits locally at every place, then it splits globally, which means it is trivial. From this they deduced their main theorem, that every such algebra over an algebraic number field splits over a cyclic cyclotomic extension. Together these results make it possible to classify all finite-dimensional central division algebras over a given number field.
Noether carried the work further. A subsequent paper showed that all maximal subfields of a division algebra are splitting fields, and it contains what is now called the Skolem–Noether theorem, which states that any two embeddings of a field extension into a finite-dimensional central simple algebra are conjugate. The related Brauer–Noether theorem characterises the splitting fields of a central division algebra. In 1932, for her fiftieth birthday, Hasse dedicated a paper to her proving a noncommutative reciprocity law, confirming her hunch that some noncommutative questions are simpler than commutative ones. That September she addressed the International Congress of Mathematicians in Zürich on hypercomplex systems and their relations to commutative algebra and number theory.
The result is a landmark in the theory of algebras. It showed Noether's abstract, conceptual methods solving concrete problems in number theory, and it remains a standard part of the theory of central simple algebras today.
Why it matters
The theorem gave number theorists a complete classification that remains a standard part of the theory of algebras.
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 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
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