Emmy Noether - Biography - Seminal Work in Abstract Algebra

Seminal Work in Abstract Algebra

Although Noether's theorem had a profound effect upon physics, among mathematicians she is best remembered for her seminal contributions to abstract algebra. As Nathan Jacobson says in his Introduction to Noether's Collected Papers,

The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her – in published papers, in lectures, and in personal influence on her contemporaries.

Noether's groundbreaking work in algebra began in 1920. In collaboration with W. Schmeidler, she then published a paper about the theory of ideals in which they defined left and right ideals in a ring. The following year she published a landmark paper called Idealtheorie in Ringbereichen, analyzing ascending chain conditions with regard to (mathematical) ideals. Noted algebraist Irving Kaplansky called this work "revolutionary"; the publication gave rise to the term "Noetherian ring", and several other mathematical objects being called Noetherian.

In 1924 a young Dutch mathematician, B. L. van der Waerden, arrived at the University of Göttingen. He immediately began working with Noether, who provided invaluable methods of abstract conceptualization. van der Waerden later said that her originality was "absolute beyond comparison". In 1931 he published Moderne Algebra, a central text in the field; its second volume borrowed heavily from Noether's work. Although Emmy Noether did not seek recognition, he included as a note in the seventh edition "based in part on lectures by E. Artin and E. Noether". She sometimes allowed her colleagues and students to receive credit for her ideas, helping them develop their careers at the expense of her own.

van der Waerden's visit was part of a convergence of mathematicians from all over the world to Göttingen, which became a major hub of mathematical and physical research. From 1926 to 1930 Russian topologist Pavel Alexandrov lectured at the university, and he and Noether quickly became good friends. He began referring to her as der Noether, using the masculine German article as a term of endearment to show his respect. She tried to arrange for him to obtain a position at Göttingen as a regular professor, but was only able to help him secure a scholarship from the Rockefeller Foundation. They met regularly and enjoyed discussions about the intersections of algebra and topology. In his 1935 memorial address, Alexandrov named Emmy Noether "the greatest woman mathematician of all time".