User:WillowW/Noether notes

Planning
To make Emmy Noether and abstract algebra into a featured topic, we need the following elements:


 * Group (mathematics)
 * Ring (mathematics)
 * Module (mathematics)
 * Algebra (ring theory)
 * Ideal (ring theory)
 * Residue class (ring theory)
 * Division ring
 * Field (mathematics)
 * Hypercomplex number
 * Group representation
 * Center (algebra)

Other possible topics include


 * Integral domain
 * Unique factorization domain
 * Principal ideal domain
 * Euclidean domain

Let's also not forget:


 * Galois group
 * Brauer group
 * Verschränktes Produkt
 * Central simple algebra

Intro
The four basic objects in abstract algebra are the group, the ring, the module and the algebra, each having its own subtypes. A group consists of one set of elements and one operation by which to combine them; the operation must be reversible (invertible) and satisfy a few other properties such as associativity. A ring likewise has one set of elements, but now has two operations by which to combine them; the first operation forms a group, but the second operation need not be invertible. If the second operation is invertible, the ring is called a division ring, and if the second operation is also commutative, the ring is called a field. A module and an algebra both have two sets of elements, and they have four and five operations, respectively, by which to combine them. In a module, the first set of elements combine among themselves as a ring (two operations), whereas those of the second set combine among themselves as a group (one operation); the module's fourth operation combines elements of the two sets together. An algebra is a module that has yet another operation by which elements of the second set can be combined; this fifth operation is the only one that may be non-associative, as in the octonions.

Words such as "element" and "combining operation" are very general, and can be applied to many real-world situations. 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. As van der Waerden recalled in his obituary of her,

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 adopted by other mathematicians and, after her death, flowered into new forms, such as category theory.

Ideal notes
It is also not always possible to divide two elements of a ring. This is true even in the integers: 1 and 3 are both integers, but 1/3 is not. Instead it is only possible to divide two numbers when their quotient has no remainder. For example, a number is evenly divisible by 3 exactly when it is a multiple of 3&mdash;that is, when it is one of ... &minus;6, &minus;3, 0, 3, 6, 9, ... The set of all multiples of 3 form an ideal of the ring of integers. Questions about divisibility in the integers can be translated to other rings by rephrasing them using ideals. The number of integers is infinite, but many rings are finite; for example, the hours displayed on a clock are the elements of a ring. The addition and multiplication in this ring are the addition and multiplication of integers carried out modulo 12, meaning that two elements are considered equal if their difference is divisible by 12. This ring has 12 elements: 0, 1, 2, 3, and so on up to 11. The element 0 is usually written 12 on a clock, but they are the same in this ring as their difference, which is 12, is divisible by 12. The elements of this ring are the 12 residue classes associated with the ideal of 12 in the ring of integers.

Ascending/descending chain conditions
Noether is famous for her deft use of ascending (Teilerkettensatz) or descending (Vielfachenkettensatz) chain conditions in her proofs. A sequence of non-empty subsets A1, A2, A3, etc. of a set S is usually said to be strictly ascending, if each is a subset of the next



A_{1} \subset A_{2} \subset A_{3} \ldots $$

The ascending chain condition requires that such sequences break off after a finite number of steps; in other words, all such sequences of subsets must be finite. Conversely, with strictly descending sequences of subsets



A_{1} \supset A_{2} \supset A_{3} \ldots $$

the descending chain condition requires that such sequences break off after a finite number.

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. However, Noether showed other mathematicians how to exploit such conditions to maximum advantage, e.g., by allowing them to conclude 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 are often crucial steps in a proof.

Several types of objects in abstract algebra have been defined with an ascending or descending chain condition, usually denoted as 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 space is a topological space in which every strictly decreasing chain of closed subspaces break off after a finite number of terms. A Noetherian module is a module in which every strictly ascending chain of submodules breaks off after a finite number.

The chain condition is often "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 may also 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 an important method in proving that a mathematical object cannot exist. Let there be a partially ordered set S in which every non-empty subset contains a minimal/maximal element. The basic idea of Noetherian induction is to show that a given subset A is such that, for all elements x in A, there must be a smaller element y. If this can be shown, then A must the empty set; its supposed elements don't exist.



Conceptual approach to mathematics
Noether is also famous for her extremely abstract approach to mathematics, which was uncommon in her time. vdWaerden quote "Begriffliche Mathematik"

Ideals
use integers to explain concepts throughout

explain Dedekind solution to deal with non-unique prime factorization in algebraic number fields: replace relations of individual numbers, e.g., 33 = 3*11, with an abstract multiplication of sets (a)(b) = (c)

mention Noether's respect for Dedekind (Es steht alles schon bei Dedekind), but also explain how she extended his work, beginning with left and right ideals

Sundries
E. Fischer arrived in Erlangen in 1911

Many mathematicians consider ''Ideale in Ringbereichen" to be her most important paper (Women in Mathematics, p. 168)

E. Noether's system of ternary, biquadratic forms had 331 explicitly written out covariant forms.

Noether's theorem is rather atypical of her work, being more calculus based and not algebraic/symbolic

have we mentioned her editing of Dedekind's Collected Works (1930–1932) with Robert Fricke and Oystein Ore, and of the letters btwn. Cantor and Dedekind, with Jean Cavaillès?

have we mentioned that she helped edit the Mathematische Annalen?

Need to add list of doctoral students supervised by her, with their thesis titles and wikilinks if any

More discussion of her driving Alexandrov and Hopf to incorporate group theory into combinatorial topology, helping develop algebraic topology

(Women in Mathematics, p. 167) mathematician Jean Dieudonné wrote in a review of her Collected Works (published in 1983 by Springer Verlag) that Noether was "by far the best woman mathematician of all time, and one of the greatest mathematicains (male or female) of the XXth century."

Pacifism
Noether was a committed pacifist for her whole life. Pavel Alexandrov writes in obituary of her, "with all her being she hated war and chauvinism in all of its manifestations", but

...her kindness and gentleness never made her weak or unable to resist evil. She had her opinions and was able to advance them with great force and persistence. Though mild and forgiving, her nature was also passionate, tempermental, and strong-willed; she always stated her opinions forthrightly, and did not fear objections.

Feminism
According to Olga Taussky-Todd, Noether was "not uninterested in the many problems women face." Attended meeting of International Federation of University Women (IFUW), and expressed opinion that one ought to go to such meetings.

Did not resent her earlier struggles

In favor of career women marrying, without thinking through the consequences

She and Ilse Brauer tried to marry off all four [Taussky, Ruth Stauffer (McKee), Grace Shover (Quin), Marie Weiss], but failed.

Gave young men preference in her recommendations for jobs so that they could start families

Devotion to students
"But she took extreme offense and sharply protested when the least injustice was done to one of her students."

Contributions of Fritz
Noetherian integral equation and Noetherian operator


 * ref: Encycl. Math., pp. pp. 409–411

Contributions of Max
Noether-Enriques theorem (Encycl. Math., p. 406)

Biograph. Dict., pp1871–1873.

Given two algebraic curves defined by two equations f=0 and g=0 in 2D (x and y) that intersect in a finite number of points, then any algebraic variety that passes through all of those intersection points can be expressed as Af + Bg = 0 (where A and B are polynomials in x and y) if and only if certain Noetherian conditions are satisfied. Extended to surfaces and hypersurfaces by Julius Koenig in 1903.

Related to ideals by Lasker and then Emmy

relate the fact that [a curve can be expressed as the intersection of two varieties in many ways, e.g., circle from 2 spheres, or circle from cone and plane)] to the non-unique factorization of algebraic numbers. Solution is not to look at this or that polynomial but rather the set of all polynomial equations that  are satisfied by the points of the intersection: use ideals.  Polynomial ideal of A1 f1 + A2 f2 + ... satisfy ring and ideal conditions

Mathematical subjects named after Emmy Noether

 * Noether's theorem


 * Noetherian
 * Noetherian group
 * Noetherian ring
 * Noetherian module
 * Noetherian space
 * Noetherian induction
 * Noetherian scheme
 * Noether normalization lemma
 * Noether problem
 * Lasker–Noether theorem
 * Skolem–Noether theorem
 * Brauer–Noether theorem



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. A common joke is that a topologist can't distinguish her donut from her coffee mug, since they can be smoothly deformed into one another.

Noether is credited with the 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 Pavel Alexandrov, Noether attended lectures given by Heinz Hopf and himself in the summers of 1926 and 1927, where "she continually made observations, which were often deep and subtle". He continues

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 simple to understand, and Hopf's own work on this subject, Eine Verallgemeinerung der Euler-Poincaréschen Formel (A Generalization of the Euler–Poincaré Formula, 1928) "bears the imprint of these remarks of Emmy Noether". Noether herself mentions her topology ideas only as an aside in one 1926 publication, where she cites it as an application of group theory.

The algebraic approach to topology was 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.

Glossary of Emmy Noether's mathematics
make collapsible template with sundry definitions

also maybe make a tree diagram to illustrate hierarchy of ring, division ring, field, etc.?

Personal "to do"
Add van der Waerden's charactizations in the obituary to list of publications by Emmy Noether, plus her own charactizations from her habilitation application CV

Ideas for illustrating algebraic objects
Need to find good everyday (non-numerical) examples of various objects in abstract algebra

Non-commutative group: Dolls and dresses
Suppose that you get a set of dolls (A, B, C,...), each with their own dresses (a, b, c,...). At any one time, you can hold only one doll in your hand. The way to play the game is to put dress x onto doll Y. For example, to put dress b on doll A, you could write



(Aa)*(Bb) = (Ab)

$$

It's a non-commutative group! :) It can be checked that it's associative.

Non-commutative ring: Rotating and recombining photographs
Suppose that you're given a set of photographs of faces, each divided into the top (above the nose) and bottom (below the nose). So the photographs of the initial faces can be symbolized as (Aa), (Bb), ... The additive operation in a ring must be an abelian group, so let's choose that to be rotating by increments of 90 degrees, a cyclic group with four members. For the multiplicative operation, let's choose to swap the bottom part of the face as before, (Aa)(Ab) = (Ab) means putting the top of face A with the bottom of face b.