Amalie Emmy Noether’s innovations in higher algebra gained her recognition as the most creative abstract algebraist of modern times. “The final formulation of “modern algebra” was due to Emmy Noether and her school.”(1) Generations of 20th-century mathematicians learned this subject from Noether’s student, van der Waerden’s famous Modern Algebra. Noether is considered the greatest female mathematician in history.

 

Bio

 

Noether received a Ph.D. from the University of Erlangen in 1907, with a dissertation on algebraic invariants. From 1913, she lectured occasionally at Erlangen, substituting for her father, Max Noether (1844–1921). In 1915, she went to the University of Göttingen and was persuaded by the eminent mathematicians David Hilbert and Felix Klein to remain there, over the objections of some faculty members. She won formal admission as an academic lecturer in 1919. While teaching under Hilbert, she provided an elegant pure mathematical formulation for several concepts of the general theory of relativity. When the Nazis came to power in Germany in 1933, Noether and many other Jewish professors at Göttingen were dismissed. In October, she left for the United States to become visiting professor of mathematics at Byrn Mawr College and to lecture and conduct research at the Institute for Advanced Study, Princeton, New Jersey.

 

Mathematics–Abstract Algebra

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Her most important contributions were in the area of abstract algebra. Algebra had changed considerably in the nearly 1,500 years that intervened between Hypatia’s Diophantine Analysis and Noether’s day. Where Diophantus and Hypatia had been concerned with equation solving and the results of algebraic operations, modern practitioners were more concerned with the formal properties of the algebraic operations. Twentieth-century abstract algebraists were studying commutativity, associativity, and distributivity. They were investigating the mathematical systems that result if one of these laws is not assumed; and they were generalizing the number system to other systems, which had been named fields, rings, groups, near rings, and so forth. This axiomatic trend had been accelerated by the work of such people as Leopold Kronecker, G. Peano, K. Weierstrass and S. Kovalevskaya, J. W. R. Dedekind, and D. Hilbert, among others.

 

The appearance of Bertrand Russell’s Principia Mathematica in 1910–13 introduced new controversies and added totally new logics to the vast domain of mathematics. Just as Newton’s Principia had established a novel way of thinking for decades after its publication, so did Russell’s Principia help to establish research into symbolic logic as a major new branch of mathematics. The enormously expanded role of the axiomatic approach became perhaps the most remarkable aspect of twentieth-century mathematics.(2)

 

Emmy Noether was close to the development of this trend. She had built on her father’s work, particularly his residual theorem; during the 1920s, she fitted this theorem into her general theory of ideals in arbitrary rings, helping to further establish the axiomatic and integrative tendencies of abstract algebra.

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What are rings and fields in mathematics?  

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Arithmetic with integers involves two fundamental operations: addition and multiplication (after which one learns about subtraction and division as well). The sum 1 + 4 + 9 + 16 requires no parentheses because you can start anywhere in this sum, even rearranging the terms, and always get the same answer (because addition is associative and commutative). You learn the distributive property of integers: a x (b + c) = a x b + a x c. Many sets possess these same useful properties exhibited by integers. They are not listed here, but you give all sets with these properties a name: rings. The set of real numbers is also a ring, although it has an additional useful property integers do not possess. With integers, although you can add or multiple two integers and obtain an integer, and you can also subtract two integers and get an integer, you cannot necessarily divide two integers and get an integer. On the other hand, you can divide any real number by another other real number (other than zero!) and get a real number. This distinction gives the set of real numbers the designation of field.

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Rings and fields were historically important because they allowed mathematicians to translate some classical problems into a brand-new language.(3) This new language allowed for long-desired proofs that the circle cannot be squared, nor the cube doubled, nor an arbitrary angle trisected using only straightedge and compass. It also allowed mathematicians to prove that in spite of the existence of a quadratic formula--and cubic and quartic formulas--no such formula could exist for quintic (i.e. 5th degree) polynomials.

 

Noether’s Theorem in Quantum Physics

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This powerful principle is a foundation stone of quantum physics. One of the great intellectual achievements of the twentieth century, quantum physics describes the behavior of atoms, nuclei, and subnuclear particles. Noether’s theorem proves that fundamental laws about the conservation of energy, momentum, angular momentum, and so on, are identical with the laws of symmetries. As a result, the laws of physics are independent of time or place; they hold true next year or in a thousand years, and a problem has the same outcome no matter where it is done. As Albert Einstein wrote to David Hilbert, “Yesterday I received from Miss Noether a very interesting paper on invariant forms. It amazed me that one could view these things from such a general standpoint. It would not have done the Old Guard at Göttingen any harm, had they picked up a thing or two from her. She certainly knows what she is doing.”(4)

 

Noether’s place as a mathematician can best be judged by other mathematicians. Einstein said of her, “In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.”(5) Hermann Weyl, an old friend who delivered her eulogy, said, “She was a great mathematician, the greatest, I firmly believe, that her sex has ever produced.”(6)

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Footnotes:

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(1) Isabella Bashmakova and Galina Smirnova, translated from the Russian by Abe Shenitzer, The Beginnings & Evolution of Algebra (Washingon, D.C., 2000), p. 144.

(2) Lynn M. Osen, Women in Mathematics,   (Cambridge, 1974), p. 147.

(3) Richard Brown, editor, 30-Second Math - The 50 Most Mind-Expanding Theories In Mathematics, Each Explained In Half a Minute (London, 2012), p. 88. 

(4) Sharon Bertsch McGrayne, Nobel Prize Women in Science – Their Lives, Struggles and Momentous Discoveries (New York, 1993), p. 73.

(5) Ibid., p. 151.

(6) Ibid., p. 152.