# nLab Emmy Noether

Emmy Noether (1882–1935) was a German mathematician with important results in theoretical physics (see Noether's theorem) and abstract algebra, and whose massive influence on algebra in the 20th century was spread, in part, through Bartel Leendert van der Waerden?‘s influential text Moderne Algebra, based on her ideas. David Hilbert fought to have her allowed to teach at Göttingen at a time when women had barely been allowed to attend university as students. Albert Einstein wrote, on her death,

In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.

Noether gave a plenary talk at the 1932 International Congress of Mathematicians

From the nPOV, one of the most important contributions Noether made was the categorification of the Betti numbers and torsion coefficients of a topological space to homology groups, and the implicit understanding that homology was a functor. The following is the abstract of her talk at the DMV in 1925 where she pointed out this fundamental result

Ableitung der Elementarteilertheorie aus der Gruppentheorie. Die Elementarteilertheorie gibt bekanntlich für Moduln aus ganzzahligen Linearformen eine Normalbasis von der Form $(e_1y_1, e_2y_2, ..., e_ry_r)$, wo jedes $e$ durch das folgende teilbar ist; die $e$ sind dadurch bis aufs Vorzeichen eindeutig festgelegt. Da jede Abelsche Gruppe mit endlich vielen Erzeugenden dem Restklassensystem nach einem solchen Modul isomorph ist, ist dadurch der Zerlegungssatz dieser Gruppen als direkte Summe größter zyklischer mitbewiesen. Es wird nun umgekehrt der Zerlegungssatz rein gruppentheoretisch direkt gewonnen, in Verallgemeinerung des für endliche Gruppen üblichen Beweises, und daraus durch Übergang vom Restklassensystem zum Modul selbst die Elementarteilertheorie abgeleitet. Der Gruppensatz erweist sich so als der einfachere Satz; in den Anwendungen des Gruppensatzes — z.B. Bettische und Torsionszahlen in der Topologie — ist somit ein Zurückgehen auf die Elementarteilertheorie nicht erforderlich. -Jahresbericht der Deutschen Mathematiker-Vereinigung 34 pt 2 (1926) p 104 (source)

## Family

Noether is one of several mathematicians in the family with results named after them: her father Max was also a mathematician, and made contributions to classical 19th century algebraic geometry; the Brill-Noether theorem? is named for Max Noether, not Emmy Noether. Her brother Fritz is also known for work in applied mathematics, and contributed to the Herglotz-Noether theorem?.

## References

Extensive information is available at

and

An annotated bibliography is available on Wikipedia:

The following mid-career paper returns to a topic related to Noether’s doctoral work (under Paul Gordan?), and gave a constructive approach to a problem that previously used the mere existence result ensured Hilbert's basis theorem

category: people

Last revised on November 4, 2019 at 02:12:49. See the history of this page for a list of all contributions to it.