Editing Emmy Noether (section)

===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}}