Third epoch
(1927–1935): Hypercomplex numbers and representation theory
Much work on hypercomplex numbers and group representations
was carried out in the nineteenth and early twentieth centuries, but remained
disparate. Noether united these results and gave the first general
representation theory of groups and algebras.
Briefly, Noether subsumed the structure theory of
associative algebras and the representation theory of groups into a single
arithmetic theory of modules and ideals in rings satisfying ascending chain
conditions. This single work by Noether was of fundamental importance for the
development of modern algebra.
Third epoch
(1927–1935): Noncommutative algebra
Noether also was responsible for a number of other advances
in the field of algebra. With Emil Artin, Richard Brauer, and Helmut Hasse, she
founded the theory of central simple algebras.
A paper by Noether, Helmut Hasse, and Richard Brauer pertains
to division algebras, which are algebraic systems in which division is
possible. They proved two important theorems: a local-global theorem stating
that if a finite-dimensional central division algebra over a number field
splits locally everywhere then it splits globally (so is trivial), and from
this, deduced their Hauptsatz ("main theorem"):
“Every finite
dimensional central division algebra over an algebraic number field F splits
over a cyclic cyclotomic extension.”
These theorems allow one to classify all finite-dimensional
central division algebras over a given number field. A subsequent paper by
Noether showed, as a special case of a more general theorem, that all maximal
subfields of a division algebra D are splitting fields. This paper also contains the Skolem–Noether
theorem which states that any two embeddings of an extension of a field k into
a finite-dimensional central simple algebra over k, are conjugate. The
Brauer–Noether theorem gives a characterization of the splitting fields of central
division algebra over a field.
Assessment,
recognition, and memorials
Noether's work continues to be relevant for the development
of theoretical physics and mathematics and she is consistently ranked as one of
the greatest mathematicians of the twentieth century. In his obituary, fellow
algebraist BL van der Waerden says that her mathematical originality was
"absolute beyond comparison", and Hermann Weyl said that Noether
"changed the face of algebra by her work". During her lifetime and even until today,
Noether has been characterized as the greatest woman mathematician in recorded
history by mathematicians such as Pavel Alexandrov, Hermann Weyl, and Jean
Dieudonné.
In a letter to The New York Times, Albert Einstein wrote:
In the judgment 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
enormous importance in the development of the present-day younger generation of
mathematicians.
On 2 January 1935, a few months before her death, mathematician
Norbert Wiener wrote that
Miss Noether is ...
the greatest woman mathematician who has ever lived; and the greatest woman
scientist of any sort now living, and a scholar at least on the plane of Madame
Curie.
At an exhibition at the 1964 World's Fair devoted to Modern
Mathematicians, Noether was the only woman represented among the notable
mathematicians of the modern world.
Noether has been honored in several memorials,
The Association for Women in Mathematics holds a Noether
Lecture to honor women in mathematics every year; in its 2005 pamphlet for the
event, the Association characterizes Noether as "one of the great mathematicians of her time, someone who worked
and struggled for what she loved and believed in. Her life and work remain a
tremendous inspiration".
Consistent with her dedication to her students, the
University of Siegen houses its mathematics and physics departments in
buildings on the Emmy Noether Campus.
The German Research Foundation (Deutsche
Forschungsgemeinschaft) operates the Emmy Noether Programme, providing funding
to early-career researchers to rapidly qualify for a leading position in
science and research by leading an independent junior research group.
A Street in her hometown, Erlangen, has been named after
Emmy Noether and her father, Max Noether.
The successor to the secondary school she attended in
Erlangen has been renamed as the Emmy Noether School.
A series of high school workshops and competitions are held
in her honor in May of each year since 2001, originally hosted by a subsequent
woman mathematics Privatdozent of the University of Göttingen.
Perimeter Institute
for Theoretical Physics annually awards Emmy Noether Visiting Fellowships to
outstanding female theoretical physicists. Perimeter Institute is also home to
the Emmy Noether Council, a group of volunteers made up of international
community, corporate and philanthropic leaders work together to increase the
number of women in physics and mathematical physics at Perimeter Institute.
The Emmy Noether Mathematics Institute in Algebra, Geometry
and Function Theory in the Department of Mathematics and Computer Science,
Bar-Ilan University, Ramat Gan, Israel was jointly founded in 1992 by the
university, the German government and the Minerva Foundation with the aim to
stimulate research in the above fields and to encourage collaborations with
Germany. Its main topics are Algebraic Geometry, Group theory and Complex
Function Theory. Its activities include local research projects, conferences,
short-term visitors, post-doc fellowships, and the Emmy Noether lectures (an
annual series of distinguished lectures). ENI is a member of ERCOM:
"European Research Centers of Mathematics".
In 2013, The European Physical Society established the Emmy
Noether Distinction for Women in Physics. Winners have included Dr Catalina Curceanu,
Prof Sibylle Günter and Prof Anne L'Huillier.
In fiction, Emmy Nutter, the physics professor in "The
God Patent" by Ransom Stephens, is based on Emmy Noether.
Farther from home,
The crater Nöther on the far side of the Moon is named after
her.
The minor planet 7001 Noether is named for Emmy Noether.
Google put a memorial doodle created by Google artist Sophie
Diao on Google's homepage in many countries on 23 March 2015 to celebrate Emmy
Noether's 133rd birthday.
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