Editing Emmy Noether (section)

==Contributions to mathematics and physics==
Noether's work in [[abstract algebra]] and [[topology]] was influential in mathematics, while [[Noether's theorem]] has widespread consequences for [[theoretical physics]] and [[dynamical system]]s. Noether showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways.{{Sfn |Kimberling|1981|pp=11–12}} Her friend and colleague [[Hermann Weyl]] described her scholarly output in three epochs:
{{Blockquote |(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|{{Harvnb |Weyl| 1935}}}}

In the first epoch (1907–1919), Noether dealt primarily with [[invariant theory|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]]. Shortly after moving to Göttingen in 1915, she proved the two [[Noether's theorem]]s, "one of the most important mathematical theorems ever proved in guiding the development of modern physics".{{Sfn|Lederman|Hill|2004|p=73}}

In the second epoch (1920–1926), Noether devoted herself to developing the theory of [[ring (mathematics)|mathematical rings]].{{Sfn |Gilmer|1981|p= 131}} In the third epoch (1927–1935), Noether focused on [[noncommutative algebra]], [[linear map|linear transformations]], and commutative number fields.{{Sfn |Kimberling|1981|pp= 10–23}} Although the results of Noether's first epoch were impressive and useful, her fame among mathematicians rests more on the groundbreaking work she did in her second and third epochs, as noted by Hermann Weyl and B. L. van der Waerden in their obituaries of her.{{sfn|Weyl|1935}}{{Sfn|van der Waerden|1935}}

In these epochs, she was not merely applying ideas and methods of the earlier mathematicians; rather, she was crafting new systems of mathematical definitions that would be used by future mathematicians. In particular, she developed a completely new theory of [[ideal (ring theory)|ideals]] in [[ring (mathematics)|rings]], generalizing the earlier work of [[Richard Dedekind]]. She is also renowned for developing ascending chain conditions{{snd}}a simple finiteness condition that yielded powerful results in her hands.<ref name="ACC">{{harvnb|Rowe|Koreuber|2020|pp=27–30}}. See p. 27: "In 1921, Noether published her famous paper ... [which] dealt with rings whose ideals satisfy the ascending chain condition". See p. 30: "The role of chain conditions in abstract algebra begins with her now classic paper [1921] and culminates with the seminal study [1927]". See p. 28 on strong initial support for her ideas in the 1920s by Pavel Alexandrov and Helmut Hasse, despite "considerable skepticism" from French mathematicians.</ref> Such conditions and the theory of ideals enabled Noether to generalize many older results and to treat old problems from a new perspective, such as the topics of [[algebraic invariant]]s that had been studied by her father and [[elimination theory]], discussed below. 

===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 function|cubic]], [[quartic equation|quartic]], and [[quintic equation]]s, as well as on the [[root of unity|related problem]] of constructing [[regular polygon]]s using [[compass and straightedge constructions|compass and straightedge]]. Beginning with [[Carl Friedrich Gauss]]'s 1832 proof that [[prime number]]s such as five can be [[integer factorization|factored]] in [[Gaussian integer]]s,<ref>{{cite journal |first=Carl F. |last=Gauss |author-link=Carl Friedrich Gauss |title=Theoria residuorum biquadraticorum – Commentatio secunda |year=1832 |language=la |journal=Comm. Soc. Reg. Sci. Göttingen |volume=7 |pages=1–34}} Reprinted in {{cite book |title=Werke |trans-title=Complete Works of C.F. Gauss |publisher=[[Georg Olms Verlag]] |location=Hildesheim |year=1973 |pages=93–148}}</ref> [[Évariste Galois]]'s introduction of [[permutation group]]s in 1832 (although, because of his death, his papers were published only in 1846, by Liouville), [[William Rowan Hamilton]]'s description of [[quaternion]]s 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]].{{sfn|Noether|1987|p=168}}

===Background on abstract algebra and ''begriffliche Mathematik'' (conceptual mathematics)===
Two of the most basic objects in abstract algebra are [[Group (mathematics)|groups]] and [[Ring (mathematics)|rings]]:
* A ''group'' consists of a set of [[Element (mathematics)|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 [[Closure (mathematics)|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 [[associativity|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 by multiplying a number by one), and for every element there must be an [[inverse element]].{{sfn|Lang|2005|loc=II.§1|p=16}}{{sfn|Stewart|2015|pp=18–19}}
* A ''ring'' likewise, has a set of elements, but now has ''two'' operations. The first operation must make the set a [[commutativity|commutative]] group, and the second operation is [[Associative property|associative]] and [[distributivity|distributive]] with respect to the first operation. It may or may not be [[commutativity|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.{{sfn|Stewart|2015|p=182}} If every non-zero element has a [[multiplicative inverse]] (an element {{math|''x''}} such that {{math|1=''ax'' = ''xa'' = 1}}), the ring is called a ''[[division ring]]''. A ''[[field (mathematics)|field]]'' is defined as a commutative{{efn|The nomenclature is not consistent.}} division ring. For instance, the [[integer]]s form a commutative ring whose elements are the integers, and the combining operations are addition and multiplication. Any pair of integers can be [[addition|added]] or [[multiplication|multiplied]], always resulting in another integer, and the first operation, addition, is [[commutativity|commutative]], i.e., for any elements {{math|''a''}} and {{math|''b''}} in the ring, {{math|1=''a'' + ''b'' = ''b'' + ''a''}}. The second operation, multiplication, also is commutative, but that need not be true for other rings, meaning that {{math|''a''}} combined with {{math|''b''}} might be different from {{math|''b''}} combined with {{math|''a''}}. Examples of noncommutative rings include [[matrix (mathematics)|matrices]] and [[quaternion]]s. The integers do not form a division ring, because the second operation cannot always be inverted; for example, there is no integer {{math|''a''}} such that {{math|1= 3''a'' = 1}}.{{sfn|Stewart|2015|p=183}}{{sfn|Gowers et al.|2008|p=284}}

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 number]]s.{{sfn|Gowers et al.|2008|pp=699–700}} Unique factorizations do not always exist in other rings, but Noether found a unique factorization theorem, now called the [[Lasker–Noether theorem]], for the [[ideal (ring theory)|ideals]] of many rings.{{sfn|Osofsky|1994}} As detailed below, Noether's work included determining what properties ''do'' hold for all rings, devising novel analogs of the old integer theorems, and determining the minimal set of assumptions required to yield certain properties of rings.

Groups are frequently studied through ''[[group representation]]s''.{{sfn|Zee|2016|pp=89–92}} 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 describes the [[Symmetry|symmetries]] of the vector space. For example, there is a group which represents the rigid rotations of space. Rotations are a type of symmetry of space, because the laws of physics themselves do not pick out a preferred direction.{{sfn|Peres|1993|pp=215–229}} Noether used these sorts of symmetries in her work on invariants in physics.{{sfn|Zee|2016|p=180}}

A powerful way of studying rings is through their ''[[module (mathematics)|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.{{sfn|Gowers et al.|2008|p=285}}

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 an ''[[algebra over a field|algebra]]''.{{efn|The word <em>algebra</em> means both a [[algebra|subject within mathematics]] as well as an [[algebra over a field|object studied in the subject of algebra]].}} An 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.{{sfn|Lang|2002|p=121}}

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 instance, the elements might be logical propositions, where the first combining operation is [[exclusive or]] and the second is [[logical conjunction]].{{sfn|Givant|Halmos|2009|pp=14–15}} 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, van&nbsp;der&nbsp;Waerden recalled that
{{blockquote |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."{{sfn|Dick|1981|p=101}} }}

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.{{sfn|Gowers et al.|2008|p=801}}

===First epoch (1908–1919)===
====Algebraic invariant theory====
[[File:Emmy Noether - Table of invariants 2.jpg|thumb|250px|right|Table&nbsp;2 from Noether's dissertation{{Sfn|Noether|1908}} on invariant theory. This table collects 202 of the 331&nbsp;invariants of ternary biquadratic forms. These forms are graded in two variables ''x'' and ''u''. The horizontal direction of the table lists the invariants with increasing grades in ''x'', while the vertical direction lists them with increasing grades in ''u''.]]
Much of Noether's work in the first epoch of her career was associated with [[invariant theory]], principally [[algebraic invariant theory]]. Invariant theory is concerned with expressions that remain constant (invariant) under a [[group (mathematics)|group]] of transformations.{{sfn|Dieudonné|Carrell|1970}} As an everyday example, if a rigid [[metre-stick]] is rotated, the coordinates of its endpoints change, but its length remains the same.  A more sophisticated example of an ''invariant'' is the [[discriminant]] {{math|''B''<sup>2</sup> − 4''AC''}} of a homogeneous quadratic polynomial {{math|''Ax''<sup>2</sup> + ''Bxy'' + ''Cy''<sup>2</sup>}}, where {{mvar|x}} and {{mvar|y}} are [[indeterminate (variable)|indeterminate]]s. The discriminant is called "invariant" because it is not changed by linear substitutions {{math|''x'' → ''ax'' + ''by''}} and {{math|''y'' → ''cx'' + ''dy''}} with determinant {{math|1=''ad'' − ''bc'' = 1}}. These substitutions form the [[special linear group]] {{math|''SL''<sub>2</sub>}}.<ref>{{cite web|last1=Lehrer|first1=Gus|title=The fundamental theorems of invariant theory classical, quantum and super|url=https://www.math.auckland.ac.nz/~dleemans/NZMRI/lehrer.pdf|publisher=[[University of Sydney]]|access-date=9 February 2025|archive-url=https://archive.today/20250209193607/https://www.math.auckland.ac.nz/~dleemans/NZMRI/lehrer.pdf|archive-date=9 February 2025|page=8|date=January 2015|url-status=live|type=Lecture notes}}</ref>

One can ask for all polynomials in {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} that are unchanged by the action of {{math|''SL''<sub>2</sub>}}; these turn out to be the polynomials in the discriminant.{{Sfn|Schur|1968|p=45}} More generally, one can ask for the invariants of [[homogeneous polynomial]]s {{math|''A''<sub>0</sub>''x''<sup>''r''</sup>''y''<sup>0</sup> + ... + ''A<sub>r</sub>x''<sup>0</sup>''y''<sup>''r''</sup>}} of higher degree, which will be certain polynomials in the coefficients {{math|''A''<sub>0</sub>, ..., ''A<sub>r</sub>''}}, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.{{Sfn|Schur|1968}}

One of the main goals of invariant theory was to solve the "''finite basis problem''". The sum or product of any two invariants is invariant, and the finite basis problem asked whether it was possible to get all the invariants by starting with a finite list of invariants, called ''generators'', and then, adding or multiplying the generators together.{{Sfn|Reid|1996|p=30}} For example, the discriminant gives a finite basis (with one element) for the invariants of a quadratic polynomial.{{Sfn|Schur|1968|p=45}}

Noether's advisor, Paul Gordan, was known as the "king of invariant theory", and his chief contribution to mathematics was his 1870 solution of the finite basis problem for invariants of homogeneous polynomials in two variables.{{sfn |Noether|1914|p=11}}{{Sfn |Gordan| 1870}} He proved this by giving a constructive method for finding all of the invariants and their generators, but was not able to carry out this constructive approach for invariants in three or more variables. In 1890, David Hilbert proved a similar statement for the invariants of homogeneous polynomials in any number of variables.{{Sfn|Weyl|1944|pp=618–621}}{{Sfn|Hilbert|1890|p=531}} Furthermore, his method worked, not only for the special linear group, but also for some of its subgroups such as the [[special orthogonal group]].{{Sfn |Hilbert | 1890 | p = 532}}

Noether followed Gordan's lead, writing her doctoral dissertation and several other publications on invariant theory. She extended Gordan's results and also built upon Hilbert's research. Later, she would disparage this work, finding it of little interest and admitting to forgetting the details of it.{{sfn|Dick|1981|pp=16–18,155–156}} Hermann Weyl wrote, <blockquote>[A] greater contrast is hardly imaginable than between her first paper, the dissertation, and her works of maturity; for the former is an extreme example of formal computations and the latter constitute an extreme and grandiose example of conceptual axiomatic thinking in mathematics.{{sfn|Dick|1981|p=120}}</blockquote>

====Galois theory====
[[Galois theory]] concerns transformations of [[field (mathematics)|number fields]] that [[permutation|permute]] the roots of an equation.<ref>{{harvnb|Stewart|2015|pp=108–111}}</ref> Consider a polynomial equation of a variable {{math|''x''}} of [[Degree of a polynomial|degree]] {{math|''n''}}, in which the coefficients are drawn from some [[ground field]], which might be, for example, the field of [[real number]]s, [[rational number]]s, or the [[integer]]s [[modular arithmetic|modulo]]&nbsp;7. There may or may not be choices of {{math|''x''}}, which make this polynomial evaluate to zero. Such choices, if they exist, are called [[root of a function|roots]].{{sfn|Stewart|2015|pp=22-23}} For example, if the polynomial is {{math|''x''<sup>2</sup> + 1}} and the field is the real numbers, then the polynomial has no roots, because any choice of {{math|''x''}} makes the polynomial greater than or equal to one.{{sfn|Stewart|2015|pp=23, 39}} If the field is [[field extension|extended]], however, then the polynomial may gain roots,{{sfn|Stewart|2015|pp=39, 129}} and if it is extended enough, then it always has a number of roots equal to its degree.{{sfn|Stewart|2015|pp=44, 129, 148}}

Continuing the previous example, if the field is enlarged to the complex numbers, then the polynomial gains two roots, {{math|+''i''}} and {{math|−''i''}}, where {{math|''i''}} is the [[imaginary unit]], that is, {{math|1=''i''<sup> 2</sup> = −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.<ref>{{harvnb|Stewart|2015|pp=129–130}}</ref>

The [[Galois group]] of a polynomial is the set of all transformations of the splitting field which preserve the ground field and the roots of the polynomial.<ref>{{harvnb|Stewart|2015|pp=112–114}}</ref> (These transformations are called [[automorphism]]s.) The Galois group of {{nowrap|{{math|''x''<sup>2</sup> + 1}}}} consists of two elements: The identity transformation, which sends every complex number to itself, and [[complex conjugation]], which sends {{math|+''i''}} to {{math|−''i''}}. Since the Galois group does not change the ground field, it leaves the coefficients of the polynomial unchanged, so it must leave the set of all roots unchanged. Each root can move to another root, however, so transformation determines a [[permutation]] of the {{math|''n''}} roots among themselves. The significance of the Galois group derives from the [[fundamental theorem of Galois theory]], which proves that the fields lying between the ground field and the splitting field are in one-to-one correspondence with the [[subgroup]]s of the Galois group.<ref>{{harvnb|Stewart|2015|pp=114–116, 151–153}}</ref>

In 1918, Noether published a paper on the [[inverse Galois problem]].<ref>{{harvnb|Noether|1918}}.</ref> Instead of determining the Galois group of transformations of a given field and its extension, Noether asked whether, given a field and a group, it always is possible to find an extension of the field that has the given group as its Galois group. She reduced this to "[[Noether's problem]]", which asks whether the fixed field of a subgroup ''G'' of the [[symmetric group|permutation group]] {{math|''S''<sub>''n''</sub>}} acting on the field  {{math|''k''(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>)}} always is a pure [[transcendental extension]] of the field {{math|''k''}}. (She first mentioned this problem in a 1913 paper,<ref>{{harvnb|Noether|1913}}.</ref> where she attributed the problem to her colleague [[Ernst Sigismund Fischer|Fischer]].) She showed this was true for  {{math|''n''}} = 2, 3, or 4. In 1969, [[Richard Swan]] found a counter-example to Noether's problem, with  {{math|''n''}} = 47 and {{math|''G''}} a [[cyclic group]] of order&nbsp;47<ref>{{harvnb|Swan|1969|p=148}}.</ref> (although this group can be realized as a [[Galois group]] over the rationals in other ways). The inverse Galois problem remains unsolved.<ref>{{Harvnb|Malle|Matzat|1999}}.</ref>

====Physics====
{{main|Noether's theorem|Conservation law (physics)|Constant of motion}}

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]],{{sfn|Gowers et al.|2008|p=800}} 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]], in a 1918 paper.<ref>{{harvnb|Noether|1918b}}</ref> This paper presented two theorems, of which the first is known as [[Noether's theorem]].<ref>{{harvnb|Kosmann-Schwarzbach|2011|p=25}}</ref> Together, these theorems not only solve the problem for general relativity, but also determine the conserved quantities for ''every'' system of physical laws that possesses some continuous symmetry.<ref>{{cite web |last1=Lynch |first1=Peter |author-link=Peter Lynch (meteorologist) |date=18 June 2015 |title=Emmy Noether's beautiful theorem |url=https://thatsmaths.com/2015/06/18/emmy-noethers-beautiful-theorem/ |access-date=28 August 2020 |website=ThatsMaths |archive-url=https://web.archive.org/web/20231209003118/https://thatsmaths.com/2015/06/18/emmy-noethers-beautiful-theorem/ |archive-date=9 December 2023 |url-status=live}}</ref> Upon receiving her work, Einstein wrote to Hilbert:{{blockquote|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.<ref>{{Harvnb|Kimberling|1981|p=13}}</ref>}}

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.{{sfn|Zee|2016|p=180}}<ref name="ledhill">{{Harvnb|Lederman|Hill|2004|pp=97–116}}.</ref> The physical system itself need not be symmetric; a jagged asteroid tumbling in space [[Conservation of angular momentum|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 works the same way 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 law (physics)|conservation laws]] of [[momentum|linear momentum]] and [[energy]] within this system, respectively.{{sfn|Taylor|2005|pp=268–272}}<ref>{{cite book |last=Baez |first=John C. |author-link=John C. Baez |chapter=Getting to the Bottom of Noether's Theorem |pages=66–99 |title=The Philosophy and Physics of Noether's Theorems |editor-first1=James |editor-last1=Read |editor-first2=Nicholas J. |editor-last2=Teh |year=2022 |publisher=Cambridge University Press |isbn=9781108786812 |arxiv=2006.14741}}</ref>

At the time, physicists were not familiar with [[Sophus Lie]]'s theory of [[Lie group|continuous groups]], on which Noether had built. Many physicists first learned of Noether's theorem from an article by [[Edward Lee Hill]] that presented only a special case of it. Consequently, the full scope of her result was not immediately appreciated.<ref>{{harvnb|Kosmann-Schwarzbach|2011|pp=26, 101–102}}</ref> During the latter half of the 20th century, however, Noether's theorem became 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.<ref name="neeman_1999" />

===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.<ref name="ACC"/> A sequence of [[empty set|non-empty]] [[subset]]s {{math|''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>}}, ... of a [[Set (mathematics)|set]] {{math|''S''}} is usually said to be ''ascending'' if each is a subset of the next:
:<math>A_{1} \subseteq A_{2} \subseteq A_{3} \subseteq \cdots.</math>

Conversely, a sequence of subsets of {{math|''S''}} is called ''descending'' if each contains the next subset:
:<math>A_{1} \supseteq A_{2} \supseteq A_{3} \supseteq \cdots.</math>

A chain ''becomes constant after a finite number of steps'' if there is an {{math|''n''}} such that <math>A_n = A_m</math> for all {{math|''m'' ≥ ''n''}}. A collection of subsets of a given set satisfies the [[ascending chain condition]] if every 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.{{sfn|Atiyah|MacDonald|1994|p=74}} Chain conditions can be used 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.{{sfn|Atiyah|MacDonald|1994|pp=74–75}}

Many types of objects in [[abstract algebra]] can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are called ''[[Noetherian (disambiguation)|Noetherian]]'' in her honor.{{sfn|Gray|2018|p=294}} 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 (mathematics)|module]] in which every strictly ascending chain of submodules becomes constant after a finite number of steps.{{sfn|Goodearl|Warfield Jr.|2004|pp=1–3}}{{sfn|Lang|2002|pp=413–415}} A [[Noetherian space]] is a [[topological space]] whose open subsets satisfy the ascending chain condition;{{efn|Or whose closed subsets satisfy the descending chain condition.{{sfn|Hartshorne|1977|p=5}}}} this definition makes the [[spectrum of a ring|spectrum]] of a Noetherian ring a Noetherian topological space.{{sfn|Hartshorne|1977|p=5}}{{sfn|Atiyah|MacDonald|1994|p=79}}

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 Noetherian; and, ''[[mutatis mutandis]]'', the same holds for submodules and quotient modules of a Noetherian module.{{sfn|Lang|2002|p=414}} 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.{{sfn|Lang|2002|p=415–416}}

Another application of such chain conditions is in [[Noetherian induction]]{{snd}}also known as [[well-founded induction]]{{snd}}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 {{math|''S''}} is a [[partially ordered set]]. One way of proving a statement about the objects of {{math|''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 {{math|''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.<ref>{{cite web|url=https://people.engr.tamu.edu/andreas-klappenecker/cpsc289-f08/noetherian_induction.pdf|title=Noetherian induction|first=Andreas|last=Klappenecker|work=CPSC 289 Special Topics on Discrete Structures for Computing|date=Fall 2008|type=Lecture notes|publisher=[[Texas A&M University]]|access-date=14 January 2025|archive-date=4 July 2024|archive-url=https://web.archive.org/web/20240704182844/https://people.engr.tamu.edu/andreas-klappenecker/cpsc289-f08/noetherian_induction.pdf|url-status=live}}</ref>

====Commutative rings, ideals, and modules====
Noether's paper, ''Idealtheorie in Ringbereichen'' (''Theory of Ideals in Ring Domains'', 1921),{{sfn| Noether|1921}} is the foundation of general commutative [[ring theory]], and gives one of the first general definitions of a [[commutative ring]].{{efn|The first definition of an abstract ring was given by [[Abraham Fraenkel]] in 1914, but the definition in current use was initially formulated by Masazo Sono in a 1917 paper.{{sfn|Gilmer|1981|p=133}}}}{{sfn|Gilmer|1981|p=133}} 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 [[ideal (ring theory)|ideals]], every ideal is finitely generated. In 1943, French mathematician [[Claude Chevalley]] coined the term ''[[Noetherian ring]]'' to describe this property.{{sfn|Gilmer|1981|p=133}} 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.{{sfn|Rowe|Koreuber|2020|p=27}}{{sfn|Rowe|2021|p=xvi}} 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 number]]s, and that this decomposition is unique.{{sfn|Osofsky|1994}}

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)<ref>{{harvnb|Noether|1927}}.</ref> characterized the rings in which the ideals have unique factorization into prime ideals (now called [[Dedekind domain]]s).{{sfn|Noether|1983|p=13}} Noether showed that these rings were characterized by five conditions: they must satisfy the ascending and descending chain conditions, they must possess a unit element but no [[zero divisor]]s, and they must be [[integrally closed domain|integrally closed]] in their associated field of fractions.{{sfn|Noether|1983|p=13}}{{sfn|Rowe|2021|p=96}} This paper also contains what now are called the [[isomorphism theorems]],{{sfn|Rowe|2021|pp=286–287}} which describe some fundamental [[natural isomorphism]]s, and some other basic results on Noetherian and [[Artinian module]]s.{{sfn|Noether|1983|p=14}}

====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 [[polynomial factorization|factorization of polynomials]] could be carried over directly.{{sfn|Noether|1923}}{{sfn|Noether|1923b}}{{sfn|Noether|1924}} 

Traditionally, elimination theory is concerned with eliminating one or more variables from a system of polynomial equations, often by the method of [[resultant]]s.{{sfn|Cox|Little|O'Shea|2015|p=121}}
For illustration, a system of equations often can be written in the form
: {{math|1= Mv = 0 }}
where a matrix (or [[linear transform]]) {{math|M}} (without the variable {{math|x}}) times a vector {{math|v}} (that only has non-zero powers of {{math|x}}) is equal to the zero vector, {{math|0}}. Hence, the [[determinant]] of the matrix {{math|M}} must be zero, providing a new equation in which the variable {{math|x}} has been eliminated.

====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,{{Sfn | Noether| 1915}} Noether found a solution to the finite basis problem for a finite group of transformations {{math|''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 <math>\left|G\right|!</math> (the [[factorial]] of the order <math>\left|G\right|</math> of the group {{math|''G''}}). The degrees of generators need not satisfy Noether's bound when the characteristic of the field divides the number <math>\left|G\right|</math>,{{Sfn |Fleischmann | 2000 |p = 24}} but Noether was not able to determine whether this bound was correct when the characteristic of the field divides <math>\left|G\right|!</math> but not <math>\left|G\right|</math>. 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, who both showed that the bound remains true.{{Sfn |Fleischmann|2000|p=25}}{{Sfn | Fogarty |2001|p=5}}

In her 1926 paper,{{Sfn |Noether|1926}} 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 [[Haboush's theorem|Mumford conjecture]].{{sfn|Haboush|1975}} In this paper Noether also introduced the ''[[Noether normalization lemma]]'', showing that a finitely generated [[integral domain|domain]] {{math|''A''}} over a field {{math|''k''}} has a set {{math|1={''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}}} of [[algebraic independence|algebraically independent]] elements such that {{math|''A''}} is [[integrality|integral]] over {{math|1=''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>]}}.

====Topology====
[[File:Mug and Torus morph.gif|thumb|right|250px|A continuous deformation ([[homotopy]]) of a coffee cup into a doughnut ([[torus]]) and back]]
As noted by [[Hermann Weyl]] in his obituary, Noether's contributions to [[topology]] illustrate her generosity with ideas and how her insights could transform entire fields of mathematics.{{sfn|Weyl|1935}} In topology, mathematicians study the properties of objects that remain invariant even under deformation, properties such as their [[connected space|connectedness]]. An old joke is that "''a topologist cannot distinguish a donut from a coffee mug''", since they can be [[Homeomorphism|continuously deformed]] into one another.{{sfn|Hubbard|West|1991|p=204}}

Noether is credited with fundamental ideas that led to the development of [[algebraic topology]] from the earlier [[combinatorial topology]], specifically, the idea of [[Homology theory#Towards algebraic topology|homology groups]].{{Sfn |Hilton|1988|p=284}} According to Alexandrov, Noether attended lectures given by him and [[Heinz Hopf]] in the summers of 1926 and 1927, where "she continually made observations which were often deep and subtle"{{Sfn |Dick|1981|p=173}} and he continues that,
{{blockquote |When ... she first became acquainted with a systematic construction of combinatorial topology, she immediately observed that it would be worthwhile to study directly the [[group (mathematics)|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 number]]s, she suggested immediately defining the Betti group as the [[quotient group|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.{{Sfn | Dick | 1981|p= 174}}}}

Noether's suggestion that topology be studied algebraically was adopted immediately by Hopf, Alexandrov, and others,{{Sfn | Dick | 1981|p= 174}} and it became a frequent topic of discussion among the mathematicians of Göttingen.<ref>{{citation |last=Hirzebruch |first=Friedrich |author-link=Friedrich Hirzebruch |title=Emmy Noether and Topology}} in {{Harvnb | Teicher|1999|pp= 57–61}}</ref> Noether observed that her idea of a [[Betti group]] makes the [[Euler characteristic|Euler–Poincaré formula]] simpler to understand, and Hopf's own work on this subject{{Sfn |Hopf|1928}} "bears the imprint of these remarks of Emmy Noether".{{Sfn |Dick|1981|pp = 174–175}} Noether mentions her own topology ideas only as an aside in a 1926 publication,{{Sfn |Noether | 1926b}} where she cites it as an application of [[group theory]].<ref>{{citation |last=Hirzebruch |first=Friedrich |author-link=Friedrich Hirzebruch |title=Emmy Noether and Topology}} in {{Harvnb | Teicher|1999|p= 63}}</ref>

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.<ref>{{citation |last=Hirzebruch |first=Friedrich |author-link=Friedrich Hirzebruch |title=Emmy Noether and Topology}} in {{Harvnb | Teicher|1999|pp= 61–63}}</ref>

[[File:Helmut Hasse.jpg|thumb|upright|right|[[Helmut Hasse]] worked with Noether and others to found the theory of [[central simple algebra]]s.]]

===Third epoch (1927–1935)===
====Hypercomplex numbers and representation theory====
Much work on [[hypercomplex number]]s and [[group representation]]s was carried out in the nineteenth and early twentieth centuries, but remained disparate. Noether united these earlier results and gave the first general representation theory of groups and algebras.{{sfn|Noether|1929}}{{sfn|Rowe|2021|p=127}} This single work by Noether was said to have ushered in a new period in modern algebra and to have been of fundamental importance for its development.{{Sfn|van der Waerden|1985|p=244}}

Briefly, Noether subsumed the structure theory of [[associative algebra]]s and the representation theory of groups into a single arithmetic theory of [[module (mathematics)|modules]] and [[ideal (ring theory)|ideals]] in [[ring (mathematics)|rings]] satisfying [[ascending chain condition]]s.{{sfn|Rowe|2021|p=127}}

====Noncommutative algebra====
Noether also was responsible for a number of other advances in the field of algebra. With [[Emil Artin]], [[Richard Brauer]], and [[Helmut Hasse]], she founded the theory of [[central simple algebra]]s.{{Sfn |Lam | 1981 | pp= 152–153}}

A paper by Noether, Helmut Hasse, and [[Richard Brauer]] pertains to [[division algebra]]s,<ref name = "hasse_1932">{{harvnb |Brauer|Hasse|Noether|1932}}.</ref> which are algebraic systems in which division is possible. They proved two important theorems: a [[Hasse principle|local-global theorem]] stating that if a finite-dimensional central division algebra over a [[Algebraic number field|number field]] splits locally everywhere then it splits globally (so is trivial), and from this, deduced their ''Hauptsatz'' ("main theorem"):<blockquote>Every finite-dimensional [[central simple algebra|central]] [[division algebra]] over an [[algebraic number]] [[field (mathematics)|field]] F splits over a [[Abelian extension|cyclic cyclotomic extension]].</blockquote>These theorems allow one to classify all finite-dimensional central division algebras over a given number field. A subsequent paper by Noether showed, as a special case of a more general theorem, that all maximal subfields of a division algebra {{math|''D''}} are [[central simple algebra#Splitting field|splitting fields]].{{Sfn | Noether | 1933}} This paper also contains the [[Skolem–Noether theorem]], which states that any two embeddings of an extension of a field {{math|''k''}} into a finite-dimensional central simple algebra over {{math|''k''}} are conjugate. The [[Brauer–Noether theorem]]{{Sfn |Brauer | Noether | 1927}} gives a characterization of the splitting fields of a central division algebra over a field.{{sfn|Roquette|2005|p=6}}