


Emmy Noether (18821935)
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 fiftythird 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 presentday 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 4x3=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 Publishing.

