Noether

Emmy Noether (1882-1935)

Emmy Noether is one of the most significant female mathematicians in history. She was born in the Bavarian town of Erlangen. Erlangen at the time had one of Germany's three "free" Universities (i.e. independent of the churches), the other two being at Halle and Göttingen. The Erlangen University had been cast into the mathematical spotlight by one of its mathematicians named Felix Klein, who had given significant insights into the concept of a group in geometry, insights which became known as the "Erlangen Program". Emmy Noether's father, Max Noether, was a mathematician at Erlangen. He was a significant mathematician in his own right and became a Full Professor at that University.

Women were not officially allowed to study at German Universities, or to hold normal teaching positions. Nevertheless Emmy became known while enrolled as an audit student and was able eventually (in 1907) to graduate with a PhD summa cum laude at Erlangen under the supervision of Paul Gordan (whom David Hilbert had described as "King of the Invariants").

In 1915 she moved to Göttingen where she was given a licence to teach without being paid. Hilbert was in fact one of her colleagues there. Her most productive years were during the 1920s, when she produced a number of significant results. She is best known for her work in abstract algebra, particularly working with structures such as rings. She also did important work on the theory of invariants, which had an influence on the formulation of Einstein's general theory of relativity.

Also during the 1920s she spent short periods as Visiting Professor at Frankfurt and Moscow. In 1933 the Nazis withdrew her licence to teach. She left Germany and emigrated to the US, where she took up a Faculty position at Bryn Mawr, a Women's College in Pennsylvania. Bryn Mawr was not far from Princeton, where Einstein had recently arrived. Emmy Noether also gave weekly lectures there. She died suddenly on 14 April 1935 at Bryn Mawr. It is significant that Albert Einstein wrote a deeply respectful commentary which was published in the New York Times on 1 May 1935. In this commentary Einstein said

Within the past few days a distinguished mathematician, Professor Emmy Noether, formerly connected with the University of Göttingen and for the past two years at Bryn Mawr College, died in her fifty-third year. In the judgement 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 importance in the development of the present-day younger generations of mathematicians. Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relations. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper presentation into the laws of nature.

A Brief Insight into Emmy Noether's Work

Emmy Noether's results were mainly in the area of algebraic structure. Einstein (above) gives some broad ideas on the potential outcome of a knowledge of these structures. More specifically, today, a knowledge of these structures gives insight into the optimal way in which computers may be designed, computation can be performed and how data can be optimally stored.

Emmy Noether worked on such structures as ideals, rings and chains.

A ring is an abstract structure in which the objects are subject to two operations (such as addition and multiplication) and satisfy a number of axioms (rules). These axioms require the existence of certain laws, such as the associative law which must be satisfied by these operations, and the ring must include a zero element. The simplest example is the ring of integers Z, which consists of the well known numbers ...,-3,-2,-1,0,1,2,3,... (i.e. the positive and negative integers, including zero). Any two numbers can be added or multiplied (the two operations) to give a result which is also in the ring. For example 4 and -3 are two members of Z and 4x-3=-12, another member of Z.

An ideal of a ring is a subset of the ring (i.e. a structure whose elements are all in the ring), which is a ring itself, and furthermore satisfies the rule that if any element of the ideal is multiplied by any member of the ring, the result is a member of the ideal. In the case of Z the only ideals are the sets of integers divisible by a given integer. For example the ideal 2Z is the set of integers divisible by the number 2. In this case it is the set ...,-4,-2,0,2,4,... . Multiplication of any of these elements even by any number (even or odd) still leads to an even result. Also there is are ideals 3Z, 4Z, etc.

A chain is a relationship in which ideals are linked by the subset relation. For example all numbers divisible by 6 are also divisible by 3. So we can say 6Z<3Z (where we use here the symbol "<" to mean "is a subset of", rather than the normal symbol, because of font constraints) is a chain in which the first term is 6Z. In fact we would call this an ascending chain in which the first term is 6Z because each term (ideal) in the chain is a subset of the next. Since numbers divisible by 6 are also divisible by 2, this ideal also leads to the chain 6Z<2Z. Chains can be extended in length. For example we can also write 6Z<3Z<Z or 6Z<2Z<Z are both chains commencing with 6Z. Note that both of these chains cannot be further extended upwards. We say that they are finite.

Emmy Noether's name is perpetuated as the name for a ring in which every (ascending) chain of ideals is finite, as it is demonstrably in the case of Z.

The 1999 T Shirt of the Australian Mathematics Trust indeed commemorates Emmy Noether, after whom one of our Mathematics Enrichment courses is named, but also lists each of the eight chains which commence with 18Z.

18Z<9Z<3Z<Z

18Z<6Z<3Z<Z

18Z<6Z<2Z<Z

18Z<9Z<Z

18Z<6Z<Z

18Z<3Z<Z

18Z<2Z<Z

18Z<Z

Reference

Emmy Noether, Auguste Dick, Translated by HI Blocher, Birkhäuser, Basel, 1981.

Note that the photo above is reproduced from this reference with permission from the publisher. It shows Emmy Noether en route from Swinemünde to Königsberg (aboard the Steamship Danzig) to attend the annual meeting of the German Mathematical Society, September 1930. The photo was taken by her collaborator Helmut Hasse.

Written by Peter Taylor, March 1999.

This T Shirt, the finite ideal chains of 18Z, is available from the AMT.