Second epoch
(1920–1926): Commutative rings, ideals, and modules
Noether's paper, Idealtheorie in Ringbereichen (Theory of
Ideals in Ring Domains, 1921), is the foundation of general commutative ring
theory, and gives one of the first general definitions of a commutative ring. Before her paper, most results in commutative
algebra were restricted to special examples of commutative rings, such as
polynomial rings over fields or rings of algebraic integers. Noether proved
that in a ring which satisfies the ascending chain condition on ideals, every
ideal is finitely generated. In 1943, French mathematician Claude Chevalley
coined the term, Noetherian ring, to describe this property. A major result in Noether's 1921 paper is the
Lasker–Noether theorem, which extends Lasker's theorem on the primary
decomposition of ideals of polynomial rings to all Noetherian rings. The
Lasker–Noether theorem can be viewed as a generalization of the fundamental
theorem of arithmetic which states that any positive integer can be expressed
as a product of prime numbers, and that this decomposition is unique.
Noether's work Abstrakter Aufbau der Idealtheorie in
algebraischen Zahl- und Funktionenkörpern (Abstract Structure of the Theory of
Ideals in Algebraic Number and Function Fields, 1927) characterized the rings
in which the ideals have unique factorization into prime ideals as the Dedekind
domains: integral domains that are Noetherian, 0- or 1-dimensional, and
integrally closed in their quotient fields. This paper also contains what now
are called the isomorphism theorems, which describe some fundamental natural isomorphisms
and some other basic results on Noetherian and Artinian modules.
Second epoch
(1920–1926): Elimination theory
In 1923–1924, Noether applied her ideal theory to
elimination theory in a formulation that she attributed to her student, Kurt
Hentzelt. She showed that fundamental theorems about the factorization of
polynomials could be carried over directly. Traditionally, elimination theory is concerned
with eliminating one or more variables from a system of polynomial equations,
usually by the method of resultants.
For illustration, a system of equations often can be written
in the form M v = 0 where a matrix (or linear transform) M
(without the variable x) times a vector v (that only has non-zero powers
of x) is equal to the zero vector, 0. Hence, the determinant of the matrix M
must be zero, providing a new equation in which the variable x has been
eliminated.
Second epoch
(1920–1926): Invariant theory of finite groups
Techniques such as Hilbert's original non-constructive
solution to the finite basis problem could not be used to get quantitative information
about the invariants of a group action, and furthermore, they did not apply to
all group actions. In her 1915 paper, Noether found a solution to the finite
basis problem for a finite group of transformations G
acting on a finite-dimensional vector space over a field of
characteristic zero. Her solution shows that the ring of invariants is
generated by homogeneous invariants whose degree is less than, or equal to, the
order of the finite group; this is called Noether's bound. Her paper gave two proofs
of Noether's bound, both of which also work when the characteristic of the
field is coprime to |G|! (The factorial of the order |G|
of the group G). The degrees of generators need not satisfy Noether's
bound when the characteristic of the field divides the number |G| , but Noether was not able to determine
whether this bound was correct when the characteristic of the field
divides |G|! But not
|G. For many years, determining the truth or falsehood of this bound for
this particular case was an open problem, called "Noether's gap". It
was finally solved independently by Fleischmann in 2000 and Fogarty in 2001,
which both showed that the bound remains true.
In her 1926 paper, Noether extended Hilbert's theorem to
representations of a finite group over any field; the new case that did not
follow from Hilbert's work is when the characteristic of the field divides the
order of the group. Noether's result was later extended by William Haboush to
all reductive groups by his proof of the Mumford conjecture. In this paper Noether also introduced the
Noether normalization lemma, showing that a finitely generated domain A over a
field k has a set { x1, ... , xn } of algebraically independent elements such
that A is integral over k [x1, ... , xn]
.
Second epoch
(1920–1926): Contributions to topology
As noted by Pavel Alexandrov and Hermann Weyl in their
obituaries, Noether's contributions to topology illustrate her generosity with
ideas and how her insights could transform entire fields of mathematics. In
topology, mathematicians study the properties of objects that remain invariant
even under deformation, properties such as their connectedness. An old joke is
that "a topologist cannot distinguish a donut from a coffee mug",
since they can be continuously deformed into one another.
Noether is credited with fundamental ideas that led to the
development of algebraic topology from the earlier combinatorial topology,
specifically, the idea of homology groups.
According to the account of Alexandrov, Noether attended lectures given
by Heinz Hopf and by him in the summers of 1926 and 1927, where "she
continually made observations which were often deep and subtle" and he
continues that,
When ... she first
became acquainted with a systematic construction of combinatorial topology, she
immediately observed that it would be worthwhile to study directly the groups
of algebraic complexes and cycles of a given polyhedron and the subgroup of the
cycle group consisting of cycles homologous to zero; instead of the usual
definition of Betti numbers, she suggested immediately defining the Betti group
as the complementary (quotient) group of the group of all cycles by the
subgroup of cycles homologous to zero. This observation now seems self-evident.
But in those years (1925–1928) this was a completely new point of view.
Noether's suggestion that topology be studied algebraically
was adopted immediately by Hopf, Alexandrov, and others, and it became a
frequent topic of discussion among the mathematicians of Göttingen. Noether observed that her idea of a Betti
group makes the Euler–Poincaré formula simpler to understand, and Hopf's own
work on this subject "bears the imprint of these remarks of Emmy
Noether". Noether mentions her own
topology ideas only as an aside in a 1926 publication, where she cites it as an
application of group theory.
This algebraic approach to topology was also developed
independently in Austria. In a 1926–1927 course given in Vienna, Leopold
Vietoris defined a homology group, which was developed by Walther Mayer, into
an axiomatic definition in 1928.
Helmut Hasse worked with Noether and others to found the
theory of central simple algebras.
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