**KEYWORDS: **Emmy Noether, mathematics, physics, Noether’s theorem, abstract algebra, noncommutative algebra, ring theory, theory of ideals, representation theory, symmetry

**SHE THOUGHT IT**

Whilst she is not particularly well-known, especially outside scientific circles, Amelia Emmy Noether’s life path is the most accurate example of what women had to go through at the beginning of the twentieth century. Her complex work has been extremely revolutionary in the fields of mathematics and physics¹ (p. 64). She became “a founder of abstract algebra”¹ (p. 64) when she developed theories of rings and fields as well as the absolute theory of ideals.

She put forth a unified theory of non-commutative algebra and representation theory and, with colleagues Helmut Hasse and Richard Brauer, Noether deduced that each simple algebra over an algebraic number field is cyclic² (p. 2). However, her most significant contribution was to theoretical physics, which has been recognised as “certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics”³ (p. 73). The theorem, still known by her name nowadays (*Noether’s theorem*), explains the fundamental relationship between symmetry and conservation law. Put simply, every differentiable symmetry in nature has a corresponding conservation law, which implies that there is a measurable quantity that does not change in any physical process or time; or rather, a conserved quantity³ (p. 97-98). Her theorem was immediately recognised to be of fundamental importance.

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**SHORT BIOGRAPHY**

Emmy Noether was born on 23 March 1882 in Erlangen, Southern Germany, as the oldest of four children in a Jewish family. She was taught to cook and clean, as was expected of women at the time; however, she was more ambitious than that¹ (p. 67 – 68). She attended a grammar school in Erlangen, where she studied languages, piano and mathematics, and, in the late 1890s she decided to change direction and pursue a different career. Inspired by her father, Max Noether, who was a well-known professor and mathematician, she decided to study mathematics, something quite unusual for women of her days³ (p. 71). Back in 1900, women were just entering academic life, so they were only permitted to study unofficially and could only participate in some lectures with the professor’s agreement. However, she earned her bachelor’s degree three years later, becoming one of the first women to do so. In 1907, Emmy Noether earned her PhD at Göttingen University, where she attended lectures given by great mathematicians of that era. Her thesis, written under the mentorship of Paul Gordan, was focused on calculating 331 invariants of ternary biquadratic forms² (p. 5).

She returned home and spent seven years working at the Mathematical Institute in Erlangen, but without a salary. In these years, she became member of the German Association of Mathematics and took part in their annual meetings² (p.18-19). Since her reputation as a great mathematician had already spread, in 1915, Emmy Noether was invited by David Hilbert and Felix Klein to return to Göttingen to teach and join them in their mathematical department³ (p. 71). Her acceptance caused some confusion, as not everyone in Göttingen was ready to welcome a female colleague, but thanks to Hilbert’s efforts, Noether was able to work there and give her own public lectures, although still with no financial support³ (p. 72).

Very soon after her arrival, Emmy Noether proved to everyone the value of her extraordinary abilities. Her earliest work at Göttingen consisted of explaining Noether’s theorem, which states that “for every continuous symmetry in the laws of physics, there is a corresponding conservation law”³ (p. 73). The idea of symmetry was at the time a rather revolutionary way of thinking, but the importance and meaning of Noether’s theorem were recognised immediately, as well as praised by noteworthy scientists like the physicist Albert Einstein and mathematician David Hilbert³ (p. 73). Subsequently, Noether continued to focus primarily on mathematics and dedicated herself to abstract algebra; by 1919, she had already contributed to the creation of ring theory³ (p. 73). Her “Idealtheorie in Ringbereichen”, published in 1921, was considered of “foundational importance in the development of modern algebra”³ (p. 73-74). Nonetheless, it was only in 1923 that her teaching activities were finally rewarded with a meagre salary² (p. 47). Her reputation was growing and her fame spreading; between 1928 and 1932, Noether gave lectures all around the globe: at the Moscow State University and the Goethe University Frankfurt, at the International Mathematical Congress in Zurich and at the International Mathematical Congress in Bologna³ (p.73 – 74). Her successful career came to an end, however, when, in 1933, National Socialism took power in Germany. Noether, being Jewish and a woman lost her position at the University of Göttingen, along with all the other ethnic minorities there. She remained focused on mathematics and soon accepted a visiting professorship in the United States at Bryn Mawr College in 1934, but her life abroad did not last long. Emmy Noether died in 1935, at the age of 53, after a surgery to remove a large ovarian tumor³ (p. 76).

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**SCIENTIFIC COLLABORATION AND RECOGNITION**

As a classic story about the unjustifiable marginalisation of outstanding women in the scientific sphere, Emmy Noether’s life was characterised by a lot of unpaid work, unofficial positions and rejections. Despite growing up in a time and place where ethnic minorities met a tolerant and receptive academic community, studying as a woman was still an issue. Women were just starting to get permission to participate in university programmes and their presence was still rarely accepted; for instance, when working on her PhD, Emmy Noether was Paul Gordan’s only female candidate² (p. 5).

Noether became an outstanding mathematician and one of the most important investigators of abstract algebra of the twentieth century. Her work remains relevant even in contemporary mathematics and physics, yet very few know who she is. During her lifetime, she was admired and respected by some of the most important scientists in history, yet she faced sexism; she participated in changing the course of physics, yet her professorship carried no salary whatsoever; she triggered radical revisions in the very structure of mathematics, yet she remains unknown even today. For Emmy Noether, proving to be an exceptional mathematician was not enough to get the recognition she deserved.

**SHE SAID IT **

*If one proves the equality of two numbers a and b by showing first that a ≤ b and then a ≥ b, it is unfair; one should instead show that they are really equal by disclosing the inner ground for their equality.*

cited by Dick, Auguste (1981), in *Emmy Noether: 1882-1935*, Boston, Birkhauser, p. 148.

**PRIZES, ACHIEVEMENTS, HONOURS**

1932: Ackermann-Teubner Memorial Award, with Emil Artlin for their contributions to mathematics

Emmy Noether has been honoured with several memorials:

The Noether Lecture by the Association for Women in Mathematics

Emmy Noether Programme by the German Research Foundation

Emmy Noether Visiting Fellowship by Perimeter Institute for Theoretical Physics

Emmy Noether Council, group of volunteers at Perimeter Institute for Theoretical Physics

Emmy Noether Mathematics Institute in Algebra, Geometry and Function Theory, Bar-Ilan university, Israel

Several schools are named after her

**INTERTEXTUAL MATERIALS**

Chicago, Judy (1974-79), *The Dinner Party*, mixed media (art installation), Elizabeth A. Sackler Center for Feminist Art, Brooklyn Museum, NY. *Irène Joliot Curie’s name is displayed among other 999 mythical, historical and notable women on the Heritage Floor part of the installation*.

Stephens, Ransom (2009), *The God Patent*, Las Vegas, 47 North. *The physicist Emmy Nutter is based on Emmy Noether.*

Mondfrans, Jennifer (2015), *Emmy Noether,* acrylic on canvas, 46 x 61cm.

**WORKS BY EMMY NOETHER**

“Über die Bildung des Formensystems der ternären biquadratischen Form” [On Complete Systems of Invariants for Ternary Biquadratic Forms] (1908), *Journal für die reine und angewandte Mathematik, *134, pp. 23-90.

“Rationale Funktionenkörper” [Rational Function Fields] (1913), *J. Ber. D. DMV* 22, pp. 316-319.

“Der Endlichkeitssatz der Invarianten endlicher Gruppen” [The Finiteness Theorem for Invariants of Finite Groups] (1915), *Mathematische Annalen, *77, p. 89-92.

“Gleichungen mit vorgeschriebener Gruppe” [Equations with Prescribed Group] (1918), *Mathematische Annalen*, 78, pp. 221-229.

“Invariante Variationsprobleme” [Invariant Variation Problems] (1918), *Nachr. D. König. Gesellsch. D. Wiss*, pp. 235-237.

“Zur Theorie der Polynomideale und Resultanten” (1923), *Mathematische Annalen, *88, pp. 53-79.

“Eliminationstheorie und Idealtheorie” (1924), *Jahresbericht der Deutschen Mathematiker-Vereinigung, *33, pp. 116-120.

“Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Charakteristik p” [Proof of the Finiteness of the Invariants of Finite Linear Groups of Characteristic p] (1926), *Nachr. Ges. Wiss*, pp. 28-35.

“Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern” [Abstract Structure of the Theory of Ideals in Algebraic number Fields] (1927), *Mathematische Annalen, *96 (1), pp. 26-61.

with Brauer, Richard (1927), “Über minimale Zerfällungskörper irreduzibler Darstellungen” [On the Minimum Splitting Fields of Irreducible Representations] , *Sitz. Ber. D. Preuss. Akad. D. Wiss*, pp. 221-228.

“Hyperkomplexe Gröβen und Darstellungstheorie” [Hypercomplex Quantities and the Theory of Representations] (1929), *Mathematische Annalen, *30, pp. 641-692.

“Nichtkommutative Algebren” [Noncommutative Algebra] (1933), *Mathematische Zeitschrift, *37, pp. 514-541.

**FURTHER READING**

Brewer, James W. / Martha K. Smith (1982), *Emmy Noether: A Tribute to Her Life and Work*, New York, Marcel Dekker.

Byers, Nina (1996), “E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws”, *Proceedings of a Symposium on the Heritage of Emmy Noether*, IL, Bar-Ilan University.

Byers, Nina / Gary Williams (2006), *Out of the Shadows: Contributions of 20 ^{th} Century Women to Physics*, Cambridge, Cambridge University Press.

Fleischmann, Peter (2000), “The Noether bound in invariant theory of finite groups”, *Advances in Mathematics, *156 (1), pp. 23-32.

Fogarty, John (2001), “On Noether’s bound for polynomial invariants of a finite group”, *Electronic Research Announcements of the American Mathematical Society, *7 (2), pp. 5-7.

James, Ioan (2003), *Remarkable Mathematicians: From Euler to von *Neumann, Washington D.C., Mathematical Association of America.

Neuenschwander, Dwight E. (2011), *Emmy Noether’s Wonderful Theorem*, Baltimore, Maryland, Johns Hopkins University Press.

Noether, Emiliana Pasca (2005), “Emmy Noether”, in *Complexities: Women in Mathematics*, ed. Bettye Anne Case and Anne M. Leggett, New Jersey, Princeton University Press.

Noether, Gottfried E. (1987), “Emmy Noether (1882-1935)”, in *Women of Mathematics: A Bibliographic Sourcebook, *ed. Louise S. Grinstein and Paul J. Campbell, New York, Greenwood Press.

Phillips, Lee (2015), “The female mathematician who changed the course of physics – but couldn’t get a job”,* Ars Technica *<http://arstechnica.com/science/2015/05/the-female-mathematician-who-changed-the-course-of-physics-but-couldnt-get-a-job/> (last accessed 10 Jan. 2017).

Tent, M.B.W. (2008), *Emmy Noether: The Mother of Modern Algebra, *Boca Raton, FL, Taylor & Francis Group, LLC.

**WORKS CITED**

- Bertsch McGrayne, Sharon (2002),
*Nobel Prize Women in Science: Their Lives, Struggles and Momentous Discoveries*, Washington DC, Joseph Henry Press. - Dick, Auguste (1981),
*Emmy Noether: 1882-1935*, Boston, Birkhauser. - Lederman, Leon M. / Christopher T. Hill (2004),
*Symmetry and the Beautiful Universe,*New York, Prometheus Books.