First epoch
(1908–1919): Physics
Noether was brought to Göttingen in 1915 by David Hilbert
and Felix Klein, who wanted her expertise in invariant theory to help them in
understanding general relativity, a geometrical theory of gravitation developed
mainly by Albert Einstein. Hilbert had observed that the conservation of energy
seemed to be violated in general relativity, because gravitational energy could
itself gravitate. Noether provided the resolution of this paradox, and a
fundamental tool of modern theoretical physics, with Noether's first theorem,
which she proved in 1915, but did not publish until 1918. She not only solved the problem for general
relativity, but also determined the conserved quantities for every system of
physical laws that possesses some continuous symmetry. Upon receiving her work,
Einstein wrote to Hilbert:
Yesterday I received
from Miss Noether a very interesting paper on invariants. I'm impressed that
such things can be understood in such a general way. The old guard at Göttingen
should take some lessons from Miss Noether! She seems to know her stuff.
For illustration, if a physical system behaves the same,
regardless of how it is oriented in space, the physical laws that govern it are
rotationally symmetric; from this symmetry, Noether's theorem shows the angular
momentum of the system must be conserved. The physical system itself need not be
symmetric; a jagged asteroid tumbling in space conserves angular momentum
despite its asymmetry. Rather, the symmetry of the physical laws governing the
system is responsible for the conservation law. As another example, if a
physical experiment has the same outcome at any place and at any time, then its
laws are symmetric under continuous translations in space and time; by
Noether's theorem, these symmetries account for the conservation laws of linear
momentum and energy within this system, respectively.
Noether's theorem has become a fundamental tool of modern
theoretical physics, both because of the insight it gives into conservation
laws, and also, as a practical calculation tool. Her theorem allows researchers to determine
the conserved quantities from the observed symmetries of a physical system.
Conversely, it facilitates the description of a physical system based on
classes of hypothetical physical laws. For illustration, suppose that a new
physical phenomenon is discovered. Noether's theorem provides a test for
theoretical models of the phenomenon:
If the theory has a
continuous symmetry, then Noether's theorem guarantees that the theory has a
conserved quantity, and for the theory to be correct, this conservation must be
observable in experiments.
Second epoch
(1920–1926): Ascending and descending chain conditions
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:
A 1 ⊂ A 2 ⊂ A 3 ⊂
⋯ . {\displaystyle A_{1}\subset
A_{2}\subset A_{3}\subset \cdots .} A_{1} \subset A_{2} \subset A_{3} \subset
\cdots.
Conversely, a sequence of subsets of S is called descending
if each contains the next subset:
A 1 ⊃ A 2 ⊃ A 3 ⊃
⋯ . {\displaystyle A_{1}\supset
A_{2}\supset A_{3}\supset \cdots .} A_{1} \supset A_{2} \supset A_{3} \supset
\cdots.
A chain becomes constant after a finite number of steps if
there is an n such that A n = A m
{\displaystyle A_{n}=A_{m}} {\displaystyle A_{n}=A_{m}} 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.
Ascending and descending chain conditions are general,
meaning that they can be applied to many types of mathematical objects—and, on
the surface, they might not seem very powerful. Noether showed how to exploit
such conditions, however, to maximum advantage.
For example: How to use chain conditions to show that every
set of sub-objects has a maximal/minimal element or that a complex object can
be generated by a smaller number of elements. These conclusions often are
crucial steps in a proof.
Many types of objects in abstract algebra can satisfy chain
conditions, and usually if they satisfy an ascending chain condition, they are
called Noetherian in her honor. By definition, a Noetherian ring satisfies an
ascending chain condition on its left and right ideals, whereas a Noetherian
group is defined as a group in which every strictly ascending chain of
subgroups is finite. A Noetherian module is a module in which every strictly
ascending chain of submodules becomes constant after a finite number of steps.
A Noetherian space is a topological space in which every strictly ascending
chain of open subspaces becomes constant after a finite number of steps; this definition
makes the spectrum of a Noetherian ring a Noetherian topological space.
The chain condition often is "inherited" by
sub-objects. For example, all subspaces of a Noetherian space, are Noetherian
themselves; all subgroups and quotient groups of a Noetherian group are
likewise, Noetherian; and, mutatis mutandis, the same holds for submodules and
quotient modules of a Noetherian module. All quotient rings of a Noetherian
ring are Noetherian, but that does not necessarily hold for its subrings. The chain
condition also may be inherited by combinations or extensions of a Noetherian
object. For example, finite direct sums of Noetherian rings are Noetherian, as
is the ring of formal power series over a Noetherian ring.
Another application of such chain conditions is in
Noetherian induction—also known as well-founded induction—which is a
generalization of mathematical induction. It frequently is used to reduce
general statements about collections of objects to statements about specific
objects in that collection. Suppose that S is a partially ordered set. One way
of proving a statement about the objects of S is to assume the existence of a
counterexample and deduce a contradiction, thereby proving the contrapositive
of the original statement. The basic premise of Noetherian induction is that
every non-empty subset of S contains a minimal element. In particular, the set
of all counterexamples contains a minimal element, the minimal counterexample.
In order to prove the original statement, therefore, it suffices to prove
something seemingly much weaker: For any counter-example, there is a smaller
counter-example.
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