Historical context
In the century from 1832 to Noether's death in 1935, the
field of mathematics – specifically algebra – underwent a profound revolution,
whose reverberations are still being felt. Mathematicians of previous centuries
had worked on practical methods for solving specific types of equations, e.g.,
cubic, quartic, and quintic equations, as well as on the related problem of
constructing regular polygons using compass and straightedge. Beginning with
Carl Friedrich Gauss's 1832 proof that prime numbers such as five can be factored
in Gaussian integers, Évariste Galois's introduction of permutation groups in
1832 (although, because of his death, his papers were published only in 1846,
by Liouville), William Rowan Hamilton's discovery of quaternions in 1843, and
Arthur Cayley's more modern definition of groups in 1854, research turned to
determining the properties of ever-more-abstract systems defined by
ever-more-universal rules. Noether's most important contributions to
mathematics were to the development of this new field, abstract algebra.
A group consists of a set of elements and a single operation
which combines a first and a second element and returns a third. The operation
must satisfy certain constraints for it to determine a group: It must be closed
(when applied to any pair of elements of the associated set, the generated
element must also be a member of that set), it must be associative, there must
be an identity element (an element which, when combined with another element
using the operation, results in the original element, such as adding zero to a
number or multiplying it by one), and for every element there must be an
inverse element.
A ring likewise, has a set of elements, but now has two
operations. The first operation must make the set a group, and the second
operation is associative and distributive with respect to the first operation.
It may or may not be commutative; this means that the result of applying the
operation to a first and a second element is the same as to the second and
first – the order of the elements does not matter. If every non-zero element
has a multiplicative inverse (an element x such that a x = x a = 1), the ring
is called a division ring. A field is defined as a commutative division ring.
Groups are frequently studied through group representations.
In their most general form, these consist of a choice of group, a set, and an
action of the group on the set, that is, an operation which takes an element of
the group and an element of the set and returns an element of the set. Most
often, the set is a vector space, and the group represents symmetries of the
vector space. For example, there is a group which represents the rigid
rotations of space. This is a type of symmetry of space, because space itself
does not change when it is rotated even though the positions of objects in it
do. Noether used these sorts of symmetries in her work on invariants in
physics.
A powerful way of studying rings is through their modules. A
module consists of a choice of ring, another set, usually distinct from the
underlying set of the ring and called the underlying set of the module, an
operation on pairs of elements of the underlying set of the module, and an
operation which takes an element of the ring and an element of the module and
returns an element of the module.
The underlying set of the module and its operation must form
a group. A module is a ring-theoretic version of a group representation:
Ignoring the second ring operation and the operation on pairs of module
elements determines a group representation. The real utility of modules is that
the kinds of modules that exist and their interactions reveal the structure of
the ring in ways that are not apparent from the ring itself. An important
special case of this is algebra. (The
word algebra means both a subject within mathematics as well as an object
studied in the subject of algebra.) Algebra consists of a choice of two rings and
an operation which takes an element from each ring and returns an element of
the second ring. This operation makes the second ring into a module over the
first. Often the first ring is a field.
Words such as "element" and "combining
operation" are very general, and can be applied to many real-world and
abstract situations. Any set of things that obeys all the rules for one (or
two) operation(s) is, by definition, a group (or ring), and obeys all theorems
about groups (or rings). Integer numbers, and the operations of addition and
multiplication, are just one example. For example, the elements might be
computer data words, where the first combining operation is exclusive or and
the second is logical conjunction. Theorems of abstract algebra are powerful
because they are general; they govern many systems. It might be imagined that
little could be concluded about objects defined with so few properties, but
precisely therein lay Noether's gift to discover the maximum that could be
concluded from a given set of properties, or conversely, to identify the
minimum set, the essential properties responsible for a particular observation.
Unlike most mathematicians, she did not make abstractions by generalizing from
known examples; rather, she worked directly with the abstractions. In his
obituary of Noether, her student van der Waerden recalled that:
The maxim by which Emmy Noether was guided throughout her
work might be formulated as follows:
"Any
relationships between numbers, functions, and operations become transparent,
generally applicable, and fully productive only after they have been isolated
from their particular objects and been formulated as universally valid concepts."
This is the begriffliche Mathematik (purely conceptual
mathematics) that was characteristic of Noether. This style of mathematics was
consequently adopted by other mathematicians, especially in the (then new)
field of abstract algebra.
Example: Integers as
a ring
The integers form a commutative ring whose elements are the
integers, and the combining operations are addition and multiplication. Any
pair of integers can be added or multiplied, always resulting in another
integer, and the first operation, addition, is commutative, i.e., for any
elements a and b in the ring, a + b = b + a. The second operation,
multiplication, also is commutative, but that need not be true for other rings,
meaning that a combined with b might be different from b combined with a.
Examples of noncommutative rings include matrices and quaternions. The integers
do not form a division ring, because the second operation cannot always be
inverted; there is no integer a such that 3 × a = 1.
The integers have additional properties which do not generalize
to all commutative rings. An important example is the fundamental theorem of
arithmetic, which says that every positive integer can be factored uniquely
into prime numbers. Unique factorizations do not always exist in other rings,
but Noether found a unique factorization theorem, now called the Lasker–Noether
theorem, for the ideals of many rings. Much of Noether's work lay in
determining what properties do hold for all rings, in devising novel analogs of
the old integer theorems, and in determining the minimal set of assumptions
required to yield certain properties of rings.
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