Auditing Her Way In
Amalie Emmy Noether was born at Erlangen in Bavaria in 1882, the daughter of the mathematician Max Noether. She qualified as a teacher of English and French, the respectable path, then turned to mathematics at a university that did not admit women as students. She was permitted to audit lectures only with the individual consent of each professor. In 1903 she passed the matriculation examination, and in 1907 completed a doctorate on invariants under Paul Gordan, whose methods were relentlessly computational.
She came to regard that thesis as an exercise in formula manipulation rather than mathematics, and spent the following seven years at the Erlangen institute without salary or title, supervising doctoral candidates whose work was formally credited to her father or his colleagues. The arrangement was ordinary for a woman of her generation and ability. What was not ordinary was the direction her interests took, away from calculation and toward the structures that make calculation possible. She published little in those years and thought very hard about everything.
Gottingen, and a Fight Over a Title
In 1915 David Hilbert and Felix Klein invited her to Gottingen, where the new general theory of relativity had raised an awkward question about how energy is conserved in a curved spacetime. Noether answered it in 1918 with the result now called her theorem: to every continuous symmetry of a physical system there corresponds a conserved quantity. Invariance under shifts in time gives conservation of energy; invariance under shifts in position gives momentum. Modern particle physics is built on that correspondence.
The faculty nonetheless refused her the habilitation required to lecture in her own name, on the grounds that admitting a woman would disgrace the institution before returning soldiers. Hilbert argued that a university senate was not a bathing establishment and that her sex was irrelevant to the mathematics, and when the vote went against him he simply announced her courses under his own name. She was granted the habilitation in 1919 and an unofficial, poorly paid position in 1923. Even then she never held a full professorial chair in Germany.
The Architecture of Modern Algebra
Her deeper legacy is abstract algebra. Noether reframed the subject around rings, ideals and modules rather than particular equations, proved the decomposition results that underpin commutative algebra, and gave her name to the ascending chain condition and to Noetherian rings. She taught in a rapid, improvisational style that many found hard to follow and a devoted circle found transforming. Bartel van der Waerden's textbook Moderne Algebra, which shaped the field for decades, grew directly out of her lecture courses. Much of the vocabulary in which the subject is now taught is hers.
She gave ideas away without much interest in credit, and a number of results published by students and colleagues began in her seminars. In April 1933 the Nazi regime dismissed her from Gottingen for being Jewish, and she taught for a time from her flat before emigrating to Bryn Mawr College in Pennsylvania. She died there in April 1935, aged fifty-three, days after surgery for an ovarian tumour. Einstein wrote publicly that she was the most significant creative mathematical mind yet produced by the education of women.
