Born
Died
Amalie Emmy Noether
23 March 1882
Erlangen, Bavaria, Germany
14 April 1935 (aged 53)
Bryn Mawr, Pennsylvania,USA
Nationality
German
Fields
Mathematics and physics
Institutions
University of Göttingen
Bryn Mawr College
Alma mater
University of Erlangen
Doctoral advisor
Paul Gordan
Doctoral students
Max Deuring
Hans Fitting
Grete Hermann
Zeng Jiongzhi
Jacob Levitzki
Otto Schilling
Ernst Witt
Known for
Abstract algebra
Theoretical physics
Emmy's father, Max Noether, was descended from
a family of wholesale traders in Germany. He had
been paralyzed by poliomyelitis at the age of
fourteen. He regained mobility, but one leg
remained affected. Largely self-taught, he was
awarded a doctoratefrom the University of
Heidelberg in 1868. After teaching there for seven
years, he took a position in the Bavarian city
of Erlangen, where he met and married Ida Amalia
Kaufmann, the daughter of a prosperous
merchant.[8][9][10][11] Max Noether's mathematical
contributions were to algebraic geometry mainly,
following in the footsteps of Alfred Clebsch. His
best known results are the Brill–Noether
theorem and the residue, or AF+BG theorem;
several other theorems are associated with him,
including Max Noether's theorem.
First and foremost Noether is remembered by mathematicians as an algebraist and
for her work in topology. Physicists appreciate her best for her famous theorem because
of its far-ranging consequences for theoretical physics and dynamic systems. She
showed an acute propensity for abstract thought, which allowed her to approach
problems of mathematics in fresh and original ways.[86][23] Her friend and
colleague Hermann Weyl described her scholarly output in three epochs:

Emmy Noether's scientific production fell into three clearly distinct epochs:
(1) the period of relative dependence, 1907–1919;
(2) the investigations grouped around the general theory of ideals 1920–1926;

(3) the study of the non-commutative algebras, their representations by linear
transformations, and their application to the study of commutative number fields and
their arithmetics.

—Weyl 1935

In the first epoch (1907–19), Noether dealt primarily with differential and algebraic
invariants, beginning with her dissertation under Paul Gordan. Her mathematical
horizons broadened, and her work became more general and abstract, as she became
acquainted with the work of David Hilbert, through close interactions with a successor to
Gordan, Ernst Sigismund Fischer. After moving to Göttingen in 1915, she produced her
seminal work for physics, the two Noether's theorems.

In the second epoch (1920–26), Noether devoted herself to developing the theory
of mathematical rings.[87]

In the third epoch (1927–35), Noether focused on noncommutative algebra, linear
transformations, and commutative number fields.[88]

Invariant theory is concerned with expressions that remain constant
(invariant) under a group of transformations. As an everyday example, if a
rigid yardstick is rotated, the coordinates (x, y, z) of its endpoints change,
but its length L given by the formula L2 = Δx2 + Δy2 + Δz2 remains the same.
Invariant theory was an active area of research in the later nineteenth century,
prompted in part by Felix Klein's Erlangen program, according to which
different types of geometry should be characterized by their invariants under
transformations, e.g., the cross-ratio of projective geometry.
The archetypal example of an invariant is the discriminant B2 − 4AC of a
binary quadratic form Ax2 + Bxy + Cy2. This is called an invariant because it
is unchanged by linear substitutions x→ax + by,y→cx + dy with
determinant ad − bc = 1. These substitutions form the special linear
group SL2. (There are no invariants under the general linear group of all
invertible linear transformations because these transformations can be
multiplication by a scaling factor. To remedy this, classical invariant theory
also considered relative invariants, which were forms invariant up to a scale
factor.) One can ask for all polynomials in A,B, and C that are unchanged by
the action of SL2; these are called the invariants of binary quadratic forms,
and turn out to be the polynomials in the discriminant. More generally, one
can ask for the invariants of homogeneous polynomials A0xry0 + ...
+ Arx0yr of higher degree, which will be certain polynomials in the
coefficients A0, ..., Ar, and more generally still, one can ask the similar
question for homogeneous polynomials in more than two variables.

Galois theory concerns transformations of number
fields that permute the roots of an equation. Consider a
polynomial equation of a variable x of degree n, in which the
coefficients are drawn from some ground field, which might be,
for example, the field of real numbers,rational numbers, or
the integers modulo 7. There may or may not be choices of x,
which make this polynomial evaluate to zero. Such choices, if
they exist, are called roots. If the polynomial is x2 + 1 and the
field is the real numbers, then the polynomial has no roots,
because any choice of x makes the polynomial greater than or
equal to one. If the field is extended, however, then the
polynomial may gain roots, and if it is extended enough, then it
always has a number of roots equal to its degree. Continuing the
previous example, if the field is enlarged to the complex
numbers, then the polynomial gains two roots, i and −i,
where i is the imaginary unit, that is,i 2 = −1. More generally, the
extension field in which a polynomial can be factored into its
roots is known as the splitting field of the polynomial.



In this epoch, Noether became famous for her deft use of
ascending (Teilerkettensatz) or descending
(Vielfachenkettensatz) chain conditions. A sequence
of non-empty subsets A1, A2, A3, etc. of a set S is usually
said to be ascending, if each is a subset of the next
Conversely, a sequence of subsets of S is
called descending if each contains the next subset:
A chain becomes constant after a finite number of steps if
there is an n such that for all m ≥ n. A collection of
subsets of a given set satisfies the ascending chain
condition if any ascending sequence becomes constant
after a finite number of steps. It satisfies the descending
chain condition if any descending sequence becomes
constant after a finite number of steps.
Elimination theory

In 1923–24, Noether applied her ideal theory
to elimination theory—in a formulation that she
attributed to her student, Kurt Hentzelt—showing
that fundamental theorems about the factorization of
polynomials could be carried over
directly.[108][109][110] Traditionally,elimination theory is
concerned with eliminating one or more variables
from a system of polynomial equations, usually by
the method ofresultants. For illustration, the system
of equations often can be written in the form of a
matrix M (missing the variable x) times a
vectorv (having only different powers of x) equaling
the zero vector, M•v = 0. Hence, the determinant of
the matrix M must be zero, providing a new equation
in which the variable x has been eliminated.


Hypercomplex numbers and representation
theory
Noncommutative algebra


In April 1935 doctors discovered a tumor in Noether's pelvis. Worried
about complications from surgery, they ordered two days of bed rest
first. During the operation they discovered an ovarian cyst "the size of a
large cantaloupe. Two smaller tumors in her uterus appeared to be
benign and were not removed, to avoid prolonging surgery. For three
days she appeared to convalesce normally, and she recovered quickly
from a circulatory collapse on the fourth. On 14 April she fell
unconscious, her temperature soared to 109 °F (42.8 °C), and she died.
"[I]t is not easy to say what had occurred in Dr. Noether", one of the
physicians wrote. "It is possible that there was some form of unusual
and virulent infection, which struck the base of the brain where the heat
centers are supposed to be located.
A few days after Noether's death her friends and associates at Bryn Mawr
held a small memorial service at College President Park's house.
Hermann Weyl and Richard Brauer traveled from Princeton and spoke
with Wheeler and Taussky about their departed colleague. In the months
which followed, written tributes began to appear around the globe:
Albert Einstein joined Van der Waerden, Weyl, and Pavel Alexandrov in
paying their respects. Her body was cremated and the ashes interred
under the walkway around the cloisters of the M. Carey Thomas
Library at Bryn Mawr.