Skoða frumkóða fyrir Emmy Noether

[[File:Noether.jpg|thumb|Noether]]
'''Amalie Emmy Noether'''<ref name="Rufname" group="lower-alpha">[[Emmy (given name)|Emmy]] er ''[[Rufname]]'', þ.e.a.s. millinafn, en það er ætlað til frjálslegri notkunnar en fornafn eða eftirnafn. Cf. til dæmis notaði ferilskráin sem Noether sendi til Erlangen háskóla árið 1907 það (Erlangen University archive, ''Promotionsakt Emmy Noether'' (1907/08, NR. 2988); reproduced in: ''Emmy Noether, Gesammelte Abhandlungen – Collected Papers,'' ed. N. Jacobson 1983; online facsimile at [http://www.physikerinnen.de/noetherlebenslauf.html physikerinnen.de/noetherlebenslauf.html] {{Webarchive|url=https://web.archive.org/web/20070929100418/http://www.physikerinnen.de/noetherlebenslauf.html |date=2007-09-29 }}). Stundum er ''Emmy'' ruglað við styttingu á ''Amalie'', eða "Emily". e.g. {{citation|url=http://www.edge.org/documents/archive/edge52.html|author-link=Lee Smolin|first=Lee|last=Smolin|work=Edge|title=Special Relativity – Why Can't You Go Faster Than Light?|quote=Emily Noether, a great German mathematician|access-date=2019-02-26|archive-date=2012-07-30|archive-url=https://web.archive.org/web/20120730103108/http://www.edge.org/documents/archive/edge52.html|url-status=dead}}</ref> (Borið fram {{IPA link|ˈnøːtɐ}}; 23 mars 1882 – 14 apríl 1935) var þýskur [[Stærðfræði|stærðfræðingur]] sem gerði mikilvæg framlög til [[Hrein algebra|algebru]] og [[eðlisfræði]].<ref>{{cite web|url=https://www.sciencenews.org/article/emmy-noether-theorem-legacy-physics-math|title=Emmy Noether changed the face of physics; Noether linked two important concepts in physics: conservation laws and symmetries|author=Emily Conover|date=12 June 2018|website=[[Sciencenews.org]]|publisher=|access-date=2 July 2018}}</ref> Hún notaði ávallt nafnið "Emmy Noether" bæði í daglegu lífi og í skrifum. Henni var lýst af [[Pavel Alexandrov]], [[Albert Einstein]], [[Jean Dieudonné]], [[Hermann Weyl]] og [[Norbert Wiener]] sem mikilvægustu konu í sögu stærðfræðinnar.<ref name="einstein">{{Citation|last=Einstein|first=Albert|title=Professor Einstein Writes in Appreciation of a Fellow-Mathematician|date=1 May 1935|url=http://select.nytimes.com/gst/abstract.html?res=F70D1EFC3D58167A93C6A9178ED85F418385F9|publication-date=5 May 1935|newspaper=[[The New York Times]]|accessdate=13 April 2008}}. Also [http://www-history.mcs.st-andrews.ac.uk/Obits2/Noether_Emmy_Einstein.html online] at the [[MacTutor History of Mathematics archive]].</ref> Sem einn af leiðtogum stærðfræðinga sinnar tíðar, þróaði hún forsendukerfi [[Baugur (stærðfræði)|bauga]] og [[Svið (stærðfræði)|sviða]] en hún átti einnig önnur mikilvæg framlög til [[Hrein algebra|algebru]]. Kenning Noether í [[eðlisfræði]] útskýrir tengingu [[Samhverfa|samhverfna]] og [[Varðveislulögmál|varðveislulögmála]].<ref name="neeman_1999">{{Citation|author-link=Yuval Ne'eman|last=Ne'eman|first=Yuval|title=The Impact of Emmy Noether's Theorems on XXIst Century Physics}} in Teicher (1999){{Harvnb|Teicher|1999|pp=83–101}}.</ref>

Noether fæddist í [[Gyðingar|gyðingafjölskyldu]] í bænum [[Erlangen]] sem er í [[Franconia|Franconian]]<nowiki/>-héraði [[Þýskaland|Suður-Þýskalands]]; faðir hennar var stærðfræðingur, [[Max Noether]]. Upphaflega ætlaði hún sér að kenna frönsku og ensku eftir að standast þau próf sem til þurfti, henni snerist þó hugur og ákvað að læra stærðfræði í [[Háskóli Erlangen|háskóla Erlangen]] í staðinn, þar sem pabbi hennar var kennari. Eftir að hún lauk lokaritgerð sinni árið 1907 undir handleiðslu [[Paul Gordan]] þá vann hún við Stærðfræðideild Erlangen án launa í 7 ár. Á þeim tíma fengu konur yfirleitt ekki að vinna sem fræðimenn, en árið 1915 var henni boðið af [[David Hilbert]] og [[Felix Klein]] að ganga til liðs við stærðfræðideild [[Háskóli Göttingen|Göttingen-háskóla]], sem var heimsfræg miðstöð stærðfræðirannsókna. Hins vegar mótmælti heimspekideild háskólans þannig að Noether gaf út fyrirlestra undir nafni Hilberts í fjögur ár. Henni var loks leyft að gegna stöðu ''[[Privatdozent]]'' árið 1919 eftir að Hilbert lét þau orð falla að þetta væri háskóli, ekki baðhús.<ref>Haft eftir Reyni Axelsyni stærðfræðing við Háskóla Íslands.</ref>

Noether var brautryðjandi meðal meðlima stærðfræðideildar [[Göttingen]] þangað til 1933; nemendur hennar voru stundum kallaðir "Noether-drengir". Árið 1924 varð hollenski stærðfræðingurinn [[B.L. van der Waerden]] meðlimur í þeim hóp og varð fljótt talsmaður hugmynda hennar. Vinna hennar varð grunnur að öðru bindi áhrifaríkrar kennslubókar hans, ''[[Moderne Algebra]]'', sem kom út 1931. Árið 1932 ávarpaði hún [[Alþjóðlegt þing stærðfræðinga|alþjóðlegt þing stærðfræðinga]] í [[Zürich|Zurich]], en á þeim tíma var snilligáfa hennar á sviði algebru viðurkennd um allan heim. Árið eftir setti Nasisastjórn Þýskalands lög sem bönnuðu gyðingum að gegna háskólastörfum, Noether flutti þegar í stað til Bandaríkjanna þar sem hún fékk samstundis ríkisborgararétt. Þar gegndi hún stöðu við [[Bryn Mawr-háskóli|Bryn Mawr-háskólan]] í [[Pennsylvanía|Pennsylvaníu]]. Árið 1935 þurfti hún að fara í aðgerð vegna blöðru á eggjastokkum og þó svo það liti út fyrir að hún myndi jafna sig þá dó hún fjórum dögum seinna, 53 ára gömul.

Framlagi Noether til stærðfræðinnar hefur verið skipt í þrjú tímabil,<ref name="Weyl">{{Harvnb|Weyl|1935}}</ref> það fyrsta á árunum 1908-1919 þegar hún setti fram kenningar um [[algebraísk fastaskilyrði]] og [[Svið (stærðfræði)|talnasvið]]. Setning hennar um óbreytileika deildana í [[Hnikareikningar|hnikareikningum]], [[setning Noether]], hefur verið kölluð "eitt mikilvægasta og áhrifamesta framlag stærðfræðinnar til nútímaeðlisfræði". Næsta tímabil var á árunum 1920-1926, en á þeim tíma breytti hún því sem var kallað abstrakt algebra með því að búa til þau fræði sem eru grunnur fagsins nú til dags. Í klassískri fræðigrein hennar frá 1921, ''Idealtheorie in Ringbereichen'' (''Íðöl yfir bauga'') þróar Noether [[Íðal|íðöl]] í [[Baugur (stærðfræði)|víxlbaugum]] í verkfæri sem nýtast í allskyns samhengjum. Hún notfærði sér [[vaxandi keðjuskilyrði]] svo glæsilega að hlutir sem mæta þeim eru nefndir eftir henni. Þriðja tímabilið var svo á árunum 1927-1935, en þá gaf hún út greinar um [[Óvíxlin algebra|óvíxlnar algebrur]] og [[Svið (stærðfræði)|rauntalnasvið]] þar sem hún sameinaði [[Grúpa|útsetningarfræði grúpa]] og fræði [[Íðal|íðala]] og [[Mótull|mótla]]. Þó Noether væri jafn dugleg að gefa út greinar og raun ber vitni þá var hún einnig mjög gjafmild á hugmyndir sínar og er eignaður heiðurinn af mörgum fræðilegum greinum sem gefnar voru út af öðrum stærðfræðingum, þar á meðal í [[Algebraísk grannfræði|algebraískri grannfræði]] sem sótti margt í [[Svipalgebra|svipalgebru]].

== Einkalíf ==
[[Mynd:Erlangen_1916.jpg|thumb|Noether ólstu upp í Bavarian borginni Erlangen, sýnt hér á póstkorti frá 1916]]
[[Mynd:NoetherFamily_MFO3120.jpg|thumb|Emmy Noether með bræðrum sínum Alfred, Fritz, og Robert, fyrir 1918]]
Faðir Emmy, [[Max Noether]], var afkomandi fjölskyldu heildsala í Þýskalandi. Þegar hann var 14 ára lamaðist hann af [[Poliomyelitis|polio]]. Hann endurheimti hreyfigetu, en annar fótleggur hans varð þó aldrei alveg góður. Hann var að miklu leyti sjálflærður og var veitt [[doktorsgráða]] frá [[Háskólinn í Heideilberg|Háskólanum í Heidelberg]] árið 1868. Hann kenndi þar í 7 ár en samþykkti svo stöðu í [[Erlangen]] þar sem hann kynntist Idu Amalia Kaufmann, sem var dóttir vel efnaðs kaupmanns.

Framlag Max Noether til stærðfræðinnar var aðallega á sviði [[Algebraisk rúmfræði|algebraískar rúmfræði]] en hann fetaði mestmegnis í fótspor [[Alfred Clebsch]]. Best þekktu niðurstöður hans eru [[Brill–Noether setningin]] og leifin, eða [[AF+BG setningin]]; hann átti einnig aðild að nokkrum öðrum setningum.

Emmy Noether var fædd 23.&nbsp;mars 1882, elst fjögurra barna.<ref>{{cite book|title=Academic Genealogy of Mathematicians|edition=illustrated|first1=Sooyoung|last1=Chang|publisher=World Scientific|year=2011|isbn=978-981-4282-29-1|page=21|url=https://books.google.com/books?id=4siw31DPONUC}} [https://books.google.com/books?id=4siw31DPONUC&pg=PA21 Extract of p. 21]</ref> Hún var nefnd "Amalie", í höfuðið á móður sinni og föðurömmu, en byrjaði mjög ung að nota millinafnið sitt.

Fólki líkaði vel við Noether þegar hún var ung stelpa en hún skaraði ekki fram úr í skólanum. Samt sem áður var hún þekkt fyrir að vera klár og vingjarnleg. Hún var nærsýn og smámælt sem barn. Vinur fjölskyldunnar sagðist mörgum árum seinna muna eftir því að hún hefði leyst gestaþraut hratt í barnaboði. Noether lærði að elda og þrífa, eins og flestar stelpur á þessum tíma, auk þess að læra á píanó. Hún hafði ekki mikinn áhuga á neinum af þessum hlutum en fannst gaman að dansa.

Hún átti þrjá yngri bræður: Sá elsti, Alfred, var fæddur 1883, var veitt doktorsgráða í efnafræði frá Erlangen árið 1909, en dó svo níu árum seinna. Fritz Noether, fæddum 1884, er minnst fyrir afrek á sviði fræðigreina, en hann lærði í Munich og gat sér gott orð á sviði hagnýttrar stærðfræði, hins vegar flutti hann til Rússlands eftir að gyðingum var bannað að gegna háskólastöðum og var dæmdur í 25 ára fangelsi þar fyrir að vera "þýskur njósnari" og svo drepinn. Yngsti bróðir hennar Gustav Robert fæddist 1889 en lítið er vitað um hann nema að hann var langveikur og dó 1928.

== Háskólamenntun ==
[[Mynd:Paul_Albert_Gordan.jpg|thumb|Paul Gordan var leiðbeinandi Noether fyrir doktorsritgerð hennar um óbreytni fjórðastigsformúla]]
Noether var snögg að læra frönsku og ensku. Um vorið árið 1900 tók hún matspróf til að öðlast réttindi til þess að kenna þessi tungumál og fékk einkunnina ''mjög gott''. Það gaf henni réttindi til þess að kenna ensku og frönsku í stúlknaskólum, en hún kaus frekar að halda áfram að læra við háskólann í Erlangen.

Þetta var mjög óvenjuleg ákvörðun; tveimur árum áður hafði kennslustjórn háskólans lýst því yfir að það að leyfa blandað nám (kvenna og karla) myndi "henda öllu akademísku skipulagi í ringulreið". Noether var önnur tveggja kvenna í háskóla með 986 nemendum, hún mátti aðeins sækja fyrirlestra sem óreglulegur nemandi, vann sér ekki inn námseiningar og þurfti leyfi sérhvers fyrirlesara til að mega hlýða á fyrirlestra þeirra. Þrátt fyrir þessar hindranir stóðst hún lokapróf 14. júlí 1903 í ''[[Realgymnasium]]'' í [[Nuremberg]].

Á vetrarönn 1903-1904 nam hún við háskólann í Göttingen, en þar sótti hún fyrirlestra [[Karl Schwarzschild]] og stærðfræðinganna [[Hermann Minkowski]], [[Otto Blumenthal]], [[Felix Klein]] og [[David Hilbert]]. Stuttu seinna var konum formlega leyft að sækja nám í háskóla.

Noether snéri aftur til Erlangen og hóf fyrir alvöru nám í háskólanum í október 1904 undir þeirri yfirlýsingu að beina sjónum sínum eingöngu að stærðfræði. Undir handleiðslu [[Paul Gordan]] skrifaði hún svo doktorsritgerðina sína, ''Über die Bildung des Formensystems der ternären biquadratischen Form'' (''Um Fullkomin Kerfi af Föstum fyrir Þríundaforms Fjórðastigsformúlna'', 1907). Gordan var meðlimur í "reiknifræðilegum" skóla óbreytileika rannsakenda og ritgerð Noether endaði á lista með útreikningum á yfir 300 föstum. Þessi nálgun við könnun fasta var seinna leyst af hólmi með meira abstrakt og almennari aðferðum sem fundnar voru af Hilbert. Þó svo að ritgerðinni hefði verið vel tekið þá lýsti Noether henni, og öðrum svipuðum ritgerðum sínum, sem "drasli".

== Kennsla ==

=== Háskólinn í Erlangen ===
Næstu sjö ár (1908-1915) kenndi hún við stærðfræðideild háskólans í Erlangen án launa, auk þess að leysa stundum föður sinn af hólmi þegar hann var of veikur til að kenna. Árin 1910 og 1911 gaf hún út framlengingu af ritgerðinni sinni þar sem hún fór úr þremur breytum í ''n'' breytur.

Gordan fór á eftirlaun vorið 1910, en hélt samt áfram að kenna við og við ásamt eftirmanni sínum, [[Erhard Schmidt]], sem hætti stuttu seinna fyrir starf í [[Breslau]]. Gordan hætti alveg að kenna 1911 þegar eftirmaður Schmidt, [[Ernst Fischer|Ernst Fischer,]] tók við; Gordan dó svo ári seinna, í desember 1912.

[[Mynd:Emmy_noether_postcard_1915.jpg|thumb|319x319dp|Noether notaði stundum póstkort til að tala um abstrakt algebru við samstarfsmann sinn Ernst Fischer.  Þetta póstkort er frá 10 Apríl 1915.]]
Samkvæmt [[Hermann Weyl]] þá var Fischer mikilvægur áhrifavaldur Noether, þá sérstaklega með því að kynna hana fyrir framlagi [[David Hilbert]] til stærðfræðinnar. Frá 1913 til 1916 gaf Noether út þónokkrar greinar sem framlengdu og beittu niðurstöðum Hilberts á stærðfræðileg kerfi eins og [[Svið (stærðfræði)|svið]] og [[Ræð föll|ræð föll]] auk [[Fastaskilyrði|fastaskilyrða]] [[Grúpa|endanlegra grúpa]]. Þetta skeið markar byrjun rannsókna hennar á [[Hrein algebra|abstrakt algebru]], sem er það svið stærðfræðinnar sem hún unni mest.

Noether og Fischer áttu líflegt samstarf og höfðu gaman af því að iðka stærðfræði saman, þau héldu oft áfram að ræða fyrirlestra löngu eftir að þeim var lokið; Noether sendi Fischer einnig póstkort þar sem hún deildi hugleiðingum sínum um stærðfræði.

=== Háskólinn í Göttingen ===
Vorið 1915 þá buðu David Hilbert og Felix Klein Noether að koma aftur til Göttingen. Tilraunir þeirra til að fá hana til að ganga til liðs við sig voru hinsvegar hindraðar af heimspekingum og sagnfræðinum í heimspekideild skólans: Að þeirra mati ættu konur ekki að fá að vera ''privatdozenten'' sem er ólaunuð staða sem var til á þessum tíma sem virkaði þannig að nemendur þurftu að borga fyrirlesaranum fyrir að mæta á fyrirlestra. Einn af þeim lét hafa eftirfarandi eftir sér: "''Hvað munu hermenn hugsa þegar þeir koma aftur í háskóla og finna út að þeir þurfa að læra af konu?''" Hilbert svaraði þá reiðilega "''Ég sé ekki hvers vegna kyn frambjóðenda er forsenda gegn inngöngu hennar til privatdozent, við erum eftir allt saman háskóli, ekki baðhús.''"
[[Mynd:Hilbert.jpg|vinstri|thumb|Árið 1915 bauð [[David Hilbert]] Noether að vera meðlimur í stærðfræðideild Göttingen en það reyndi á viðhorf þröngsýnni samstarfsmanna hans en þeir voru alfarið á móti því að leyfa konum að kenna í háskóla.]]
Noether fór til Göttingen í Apríl en tveimur vikum seinna dó móðir hennar skyndilega í Erlangen. Hún hafði áður fengið aðhlynningu vegna augnsýkingar en ástæða hennar og áhrif þessa atburðar á Emmy eru óþekkt. Á svipuðum tíma fór faðir Noether á eftirlaun og bróðir hennar fór í þýska herinn til að taka þátt í fyrri heimsstyrjöldinni. Hún fór því aftur til Erlangen í þónokkrar vikur, aðallega til að sjá um pabba sinn.

Fyrstu árin sem hún kenndi við Göttingen þá var hún ekki með formlega stöðu og var ekki borgað; fjölskyldan hennar borgaði fyrir fæðu og húsnæði og studdu rannsóknarvinnu hennar. Fyrirlestrar hennar voru oft auglýstir undir nafni Hilberts og Noether myndi veita "aðstoð".

Það leið ekki langur tími eftir að hún kom til Göttingen þangað til hæfileikar hennar komu í ljós, hún sannaði setningu, nú þekkt sem ''[[Setning Noether]]'', sem sýnir að [[varðveislulögmál]] eru vensluð við hvaða [[Deildun|deildanlegu]] [[Samhverfa|samhverfu]] eðlisfræðilegs kerfis sem er. Greinin var kynnt af samstarfsmanni, F. Klein 26. júlí 1918 á fundi Royal Society of Sciences í Göttingen. Noether fékk ekki að vera meðlimur í þessu félagi og fékk því ekki að kynna greinina sjálf. Bandarísku eðlisfræðingarnir [[Leon M. Lederman]] og [[Cristopher T. Hill]] halda því fram í bók sinni ''Symmetry and the Beautiful Universe'' að setning Noether sé "afdráttarlaust ein mikilvægasta setning stærðfræðinnar fyrir þróun eðlisfræðinnar og sambærileg við [[Regla Pýþagórasar|reglu Pýþagórasar]]".

Þegar fyrri heimsstyrjöldin kláraðist þá olli Þýska byltingin 1918–1919 miklum breytingum í viðhorfi samfélagsins, þar á meðal bætt viðhorf gagnvart auknum réttindum kvenna. Árið 1919 leyfði loksins Háskólinn í Göttingen Noether að taka matspróf til að öðlast réttindi til að kenna við skólann. Hún tók munnlegt próf seint í maí og flutti með góðum árángri fyrirlesturinn, sem lauk prófinu, í júní 1919.

[[Mynd:Mathematik_Göttingen.jpg|thumb|210x210dp|Stærðfræðideild Göttingen Háskóla leygði Noether að taka matspróf til að öðlast réttindi til kennslu 1919, fjórum árum eftir hún hóf kennslu við skólann.]]
Þremur árum seinna þá fékk hún bréf frá Prússneska ráðherranum fyrir vísindi, list og menntun almennings, þar sem hann veitti henni titillinn ''nicht beamteter ausserordentlicher Professor'' (sem er óæviráðinn prófessor með takmarköð stjórnsýslu réttindi). Þetta var ólaunuð staða fyrir "fyrirmyndar" prófessora, ekki hær setta staðan fyrir "venjulega" prófessora, sem var  staða borguð af ríkinu. Þrátt fyrir að veita henni viðurkenningu fyrir mikilvægi rannsókna hennar þá fékk hún enn ekkert borgað. Noether fékk í raun ekki borgað fyrir neinn fyrirlestra sinna fyrr en henni var veitt sérstök staða: ''Lehrbeauftragte für Algebra'' ári seinna.

== Framlag til abstrakt algebru ==
Þó svo að setning Noether hafði veruleg áhrif á klassíska eðlisfræði og skammtafræði þá er hún meðal stærðfræðinga best þekkt fyrir framlag sitt til abstrakt algebru. Í inngangi sínum að ''Greinasafni'' (''Collected Papers'') Noether, skrifar [[Nathan Jacobson]]:<blockquote>The development of abstract algebra, which is one of the most distinctive innovations of twentieth century mathematics, is largely due to her – in published papers, in lectures, and in personal influence on her contemporaries.{{sfn|Noether|1983}}</blockquote> Hún leyfði stundum samstarfsmönnum sínum og nemendum að fá heiðurinn af hugmyndum sínum, sem hjálpaði þeim að þróa starfsferla sína á kostnað hennar eigin.

Framlag Noether til algebru byrjaði 1920. Í samstarfi við W. Schmeidler, hún gaf út grein um [[Íðal|íðöl]] þar sem þau skilgreindu vinstra og hægri íðal við [[Baugur (stærðfræði)|baug]].

Árið eftir gaf hún greinina ''Idealtheorie in Ringbereichen'', þar sem hún skoðar vaxandi keðjuskilyrði íðala í baugum. Þekktur algebrufræðingur [[Irving Kaplansky]] kallaði þessa vinnu "brautryðjandi"; þessi grein olli því að talað er um "Noether bauga" og margt annað hefur einnig verið kallað Noetherian af sömu ástæðu.

Árið 1924 var ungur hollenskur stærðfræðingur, [[Bartel Leendert van der Waerden|B.L. van der Waerden]], nemandi við Göttingen háskóla. Hann byrjaði þegar í stað að vinna með Noether, sem kenndi honum ómetanlegar aðferðir við að sjá fyrir sér abstrakt hugmyndir. Van der Waerden sagði seinna að hugvitssemi Noether "ætti sér engan samanburð". Árið 1931 gaf hann út ''Moderne Algebra'', sem er undirstöðurit í greininni; seinna bindi bókarinnar fékk heilmikið í láni frá Noether. Þó svo að Noether bað ekki um viðurkenningu þá setti hann í sjöundu útgáfu bókarinnar að "bókin væri byggð á fyrirlestrum [[Emil Artin|E. Artin]] og E. Noether".

Heimsókn Van der Waerden var partur af samleitni stærðfræðinga allstaðan að til Göttingen, sem varð þungamiðja í rannsóknum á stærðfræði og eðlisfræði. Frá 1926 til 1930 var rússneski [[Grannfræði|grannfræðingurinn]] [[Pavel Alexandrov]] fyrirlesari í skólanum en hann og Noether urðu fljótt góðir vinir. Hann byrjaði að kalla hana ''der Noether'', sem er karlkyns greinir, til þess að sýna henni virðingu. Hún reyndi að koma því í kring að hann fengi stöðu í Göttingen sem prófessor en gat bara reddað honum skólastyrk frá Rockefeller Samtökunum. Þau hittust reglulega og nutu þess að kanna skurðpunkt grannfræði og algebru. Árið 1935 í minningarræðu sinni, kallaði Alexandrov Emmy Noether "merkasta kvennkyns stærðfræðing allra tíma".

== Framhaldsnemar og áhrifaríkir fyrirlestrar ==
Noether var ekki aðeins virðingarverð fyrir stærðfræðilegt innsæi sitt heldur einnig tillitsemi gagnvart öðrum. Þó svo hún hafi stundum verið dónaleg við þá sem voru ósammála henni þá hafði hún samt sem áður orðspor fyrir viðstöðulausa hjálpsemi og þolinmæði gagnvart nýjum nemendum. Hollusta hennar við nákvæmni olli því að samstarfsmenn hennar kölluðu hana harðan gagnrýnenda en hún sameinaði þessa dómhörku við natni sem olli því að samstarfsmenn hennar lýstu henni seinna þannig:<blockquote>Completely unegotistical and free of vanity, she never claimed anything for herself, but promoted the works of her students above all.{{Sfn|van der Waerden|1935|p=98}}</blockquote>

=== Göttingen ===
[[Mynd:EmmyNoether_MFO3096.jpg|vinstri|thumb|Noether c.&nbsp;1930]]
Í Göttingen leiðbeindi Noether meira en tólf doktorsnemum; fyrsti neminn hennar var Grete Hermann, sem varði doktorsritgerðina sína í Febrúar 1925. Seinna talaði hún af lotningu um Noether sem "ritgerðar-mömmu" sína. Noether var líka leiðbeinandi [[Max Deuring]], sem var þekktur fyrir að hafa átt mikilvægt framlag til reiknings rúmfræði sem grunnemi; [[Hans Fitting]], sem er þekktur fyrir [[Setning Fitting|setningu Fitting]] og [[Hjálparsetning Fitting|hjálparsetningu Fitting]]; Zeng Jiongzhi (einnig kallaður "Chiungtze C. Tsen") sem sannaði [[Setning Tsen|setningu Tsen]]. Hún vann einnig náið með [[Wolfgang Krull]] sem átti stórt framlag í víxlinni algebru með útgáfu [[Krull's principal ideal theorem|''Hauptidealsatz'']] og jók skilning á vídd í víxlnum baugum.

Hún lifði mjög sparsamt, fyrst vegna þess hún fékk ekki borgað; en að lokum þegar hún byrjaði að fá smá borgað 1923 þá hélt hún áfram að lifa mjög einföldu lífi. Seinna fékk hún betur borgað en hún sparaði helmingin af laununum sínum til að gefa frændi, [[Gottfried E. Noether]].

Hún hafði litlar áhyggjur af útliti og mannasiðum, þeir sem hafa kynnt sér líf hennar hafa aðallega lagt áherslu á rannsóknir hennar. Frægur algebrufræðingur [[Olga Taussky-Todd]] lýsti hádegismat þar sem Noether, sem var alveg búin að gleyma sér í umræðu um stærðfræði, "veifaði höndunum í æsingi" meðan hún borðaði og "setti mat út um allt og þurrkaði sér í kjólinn sinn án nokkurra áhyggna". Nemendur hennar sem voru meðvitaðir um útlit sitt fengu aulahroll þegar hún náði í vasaklút úr blússuni sinni og lét sér fátt finnast um hárið á sér sem var í sífellt meiri flækju eftir því sem leið á fyrirlesturinn. Tveir kvennkyns nemendur komu einu sinni til hennar í hléi í tveggja klukkustunda kennslustund til þess að koma áhyggjum sínum áleiðis en voru ófærar um að koma þeim til skila vegna óstöðvandi stærðfræði umræðu sem hún var að eiga við hina nemendurna.

Samkvæmt minningargrein van der Waerden þá fylgdi hún ekki kennsluáætlun fyrir fyrirlestra sína sem fór í taugarnar á sumum nemendum. Þess í stað þá notaði hún fyrirlestrana sem vettvang til hvatvísrar umræðu um stærðfræði, þannig þau gæti hugsað um og skýrt mikilvæg vandamál í stærðfræði. Sumar af mikilvægustu niðurstöðunum hennar urðu til í þessum fyrirlestrum og fyrirlestrarnótur hennar og nemenda hennar urðu grunnurinn að mörgum mikilvægum kennslubókum, t.d. þeirra sem van der Waerden og Deuring skrifuðu.<ref>{{cite journal|author-link=Bartel Leendert van der Waerden|last=van der Waerden|first=B.L.|title=Nachruf auf Emmy Noether|trans-title=Obituary of Emmy Noether|journal=Mathematische Annalen|volume=111|year=1935|language=de|pages=469–74|doi=10.1007/BF01472233|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0111&DMDID=DMDLOG_0038&L=1|url-status=dead|archive-url=https://web.archive.org/web/20140903172427/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0111&DMDID=DMDLOG_0038&L=1|archive-date=3 September 2014|df=dmy-all}} Reprinted in {{harvnb|Dick|1981}}.</ref> Algebra er enn þann dag í dag kennd nokkurnveginn í sömu röð og Noether gerði það.

Margir samstarfsmanna hennar mættu í fyrirlestrana hennar og hún leyfði öðrum að gefa út sumar af hugmyndum sínum eins og t.d. krossfeldi tengina algebra. Noether gaf að minnsta kosti þessar fimm annar-löngu fyrirlestrarseríur:<ref name="scharlau_49">Scharlau, W. "Emmy Noether's Contributions to the Theory of Algebras" in {{Harvnb|Teicher|1999|p=49}}.</ref>

* Vetur 1924/1925: ''Gruppentheorie und hyperkomplexe Zahlen'' [''Group Theory and Hypercomplex Numbers'']
* Vetur 1927/1928: ''Hyperkomplexe Grössen und Darstellungstheorie'' [''Hypercomplex Quantities and Representation Theory'']
* Sumar 1928: ''Nichtkommutative Algebra'' [''Noncommutative Algebra'']
* Sumar 1929: ''Nichtkommutative Arithmetik'' [''Noncommutative Arithmetic'']
* Vetur 1929/30: ''Algebra der hyperkomplexen Grössen'' [''Algebra of Hypercomplex Quantities'']

Þessir kúrsar komu oft á undan mikilvægum greinum sem hún gaf út um sama efni.
<!--


Noether talaði hratt - eins og hún hugsaði - og krafðist mikillar einbeitingar frá nemendum sínum. Sumum nemendum líkaði illa við þetta sniðmát og leið eins og þeir væru ekki velkomnir. Öðrum fannst hún reiða sig of mikið á hvatvísa umræðu. Uppáhaldsnemendur hennar hinsvegar, elskuðu hvaða hún var áhugasöm og 

Noether spoke quickly – reflecting the speed of her thoughts, many said – and demanded great concentration from her students. Students who disliked her style often felt alienated.{{sfn|Mac Lane|1981|p=77}}{{Sfn|Dick|1981|p=37}} Some pupils felt that she relied too much on spontaneous discussions. Her most dedicated students, however, relished the enthusiasm with which she approached mathematics, especially since her lectures often built on earlier work they had done together.

She developed a close circle of colleagues and students who thought along similar lines and tended to exclude those who did not. "Outsiders" who occasionally visited Noether's lectures usually spent only 30&nbsp;minutes in the room before leaving in frustration or confusion. A regular student said of one such instance: "The enemy has been defeated; he has cleared out."{{sfn|Dick|1981|pp=38–41}}

Noether showed a devotion to her subject and her students that extended beyond the academic day. Once, when the building was closed for a state holiday, she gathered the class on the steps outside, led them through the woods, and lectured at a local coffee house.{{Sfn|Mac Lane|1981|p=71}} Later, after she had been dismissed by the [[Third Reich]], she invited students into her home to discuss their plans for the future and mathematical concepts.{{Sfn|Dick|1981|p=76}}

=== Moscow ===
[[Mynd:Paul_S_Alexandroff_2.jpg|thumb|205x205dp|[[Pavel Alexandrov]]]]
In the winter of 1928–1929 Noether accepted an invitation to [[Moscow State University]], where she continued working with [[Pavel Alexandrov|P.S. Alexandrov]]. In addition to carrying on with her research, she taught classes in abstract algebra and [[algebraic geometry]]. She worked with the topologists [[Lev Pontryagin]] and [[Nikolai Chebotaryov]], who later praised her contributions to the development of [[Galois theory]].{{Sfn|Dick|1981|pp=63–64}}{{Sfn|Kimberling|1981|p=26}}{{Sfn|Alexandrov|1981|pp=108–10}}

[[Mynd:Moscow_05-2012_Mokhovaya_05.jpg|hægri|thumb|Noether taught at the [[Moscow State University]] during the winter of 1928–1929.]]
Although politics was not central to her life, Noether took a keen interest in political matters and, according to Alexandrov, showed considerable support for the [[Russian Revolution]]. She was especially happy to see [[Soviet Union|Soviet]] advances in the fields of science and mathematics, which she considered indicative of new opportunities made possible by the [[Bolshevik]] project. This attitude caused her problems in Germany, culminating in her eviction from a [[Pension (lodging)|pension lodging]] building, after student leaders complained of living with "a Marxist-leaning Jewess".{{Sfn|Alexandrov|1981|pp=106–09}}

Noether planned to return to Moscow, an effort for which she received support from Alexandrov. After she left Germany in 1933 he tried to help her gain a chair at Moscow State University through the [[Narkompros|Soviet Education Ministry]]. Although this effort proved unsuccessful, they corresponded frequently during the 1930s, and in 1935 she made plans for a return to the Soviet Union.{{Sfn|Alexandrov|1981|pp=106–09}} Meanwhile, her brother [[Fritz Noether|Fritz]] accepted a position at the Research Institute for Mathematics and Mechanics in [[Tomsk]], in the Siberian Federal District of Russia, after losing his job in Germany,{{Sfn|Osen|1974|p=150}}{{Sfn|Dick|1981|pp=82–83}} and was subsequently executed during the [[Great Purge]].

== Recognition ==
In 1932 Emmy Noether and [[Emil Artin]] received the [[Ackermann–Teubner Memorial Award]] for their contributions to mathematics.<ref>{{cite web|url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Noether_Emmy.html|title=Emmy Amalie Noether|accessdate=4 September 2008|type=biography|publisher=St And.|location=UK}}</ref> The prize included a monetary reward of 500&nbsp;[[German Reichsmark|Reichsmarks]] and was seen as a long-overdue official recognition of her considerable work in the field. Nevertheless, her colleagues expressed frustration at the fact that she was not elected to the [[Göttingen Academy of Sciences|Göttingen ''Gesellschaft der Wissenschaften'']] (academy of sciences) and was never promoted to the position of ''Ordentlicher Professor''{{Sfn|Dick|1981|pp=72–73}}{{Sfn|Kimberling|1981|pp=26–27}} (full professor).{{Sfn|Dick|1981|p=188}}

[[Mynd:Zuerich_vier_Kirchen.jpg|vinstri|thumb|Noether visited [[Zürich]] in 1932 to deliver a [[List of International Congresses of Mathematicians Plenary and Invited Speakers|plenary address at the International Congress of Mathematicians]].]]
Noether's colleagues celebrated her fiftieth birthday in 1932, in typical mathematicians' style. [[Helmut Hasse]] dedicated an article to her in the ''[[Mathematische Annalen]]'', wherein he confirmed her suspicion that some aspects of [[noncommutative algebra]] are simpler than those of [[commutative algebra]], by proving a noncommutative [[Quadratic reciprocity|reciprocity law]].{{Sfn|Hasse|1933|p=731}} This pleased her immensely. He also sent her a mathematical riddle, which he called the "m<sub>μν</sub>-riddle of syllables". She solved immediately, but the riddle has been lost.{{Sfn|Dick|1981|pp=72–73}}{{Sfn|Kimberling|1981|pp=26–27}}

In November of the same year, Noether delivered a plenary address (''großer Vortrag'') on "Hyper-complex systems in their relations to commutative algebra and to number theory" at the [[International Congress of Mathematicians]] in [[Zürich]]. The congress was attended by 800&nbsp;people, including Noether's colleagues [[Hermann Weyl]], [[Edmund Landau]], and [[Wolfgang Krull]]. There were 420&nbsp;official participants and twenty-one plenary addresses presented. Apparently, Noether's prominent speaking position was a recognition of the importance of her contributions to mathematics. The 1932&nbsp;congress is sometimes described as the high point of her career.{{sfn|Kimberling|1981|pp=26–27}}{{sfn|Dick|1981|pp=74–75}}

== Expulsion from Göttingen by the Third Reich ==
When [[Adolf Hitler]] became the [[Chancellor of Germany (German Reich)|German ''Reichskanzler'']] in January&nbsp;1933, [[Nazi]] activity around the country increased dramatically. At the University of Göttingen the German Student Association led the attack on the "un-German spirit" attributed to Jews and was aided by a [[privatdozent]] named [[Werner Weber (mathematician)|Werner Weber]], a former student of Noether. [[Antisemitism|Antisemitic]] attitudes created a climate hostile to Jewish professors. One young protester reportedly demanded: "Aryan students want Aryan mathematics and not Jewish mathematics."<ref name="kim29">{{Harvnb|Kimberling|1981|p=29}}.</ref>

One of the first actions of Hitler's administration was the [[Law for the Restoration of the Professional Civil Service]] which removed Jews and politically suspect government employees (including university professors) from their jobs unless they had "demonstrated their loyalty to Germany" by serving in World War&nbsp;I. In April&nbsp;1933 Noether received a notice from the Prussian Ministry for Sciences, Art, and Public Education which read: "On the basis of paragraph&nbsp;3 of the Civil Service Code of 7&nbsp;April 1933, I hereby withdraw from you the right to teach at the University of Göttingen."{{Sfn|Dick|1981|pp=75–76}}{{sfn|Kimberling|1981|pp=28–29}} Several of Noether's colleagues, including [[Max Born]] and [[Richard Courant]], also had their positions revoked.{{sfn|Dick|1981|pp=75–76}}{{sfn|Kimberling|1981|pp=28–29}}

Noether accepted the decision calmly, providing support for others during this difficult time. [[Hermann Weyl]] later wrote that "Emmy Noether—her courage, her frankness, her unconcern about her own fate, her conciliatory spirit—was in the midst of all the hatred and meanness, despair and sorrow surrounding us, a moral solace."<ref name="kim29" /> Typically, Noether remained focused on mathematics, gathering students in her apartment to discuss [[class field theory]]. When one of her students appeared in the uniform of the Nazi [[paramilitary]] organization ''[[Sturmabteilung]]'' (SA), she showed no sign of agitation and, reportedly, even laughed about it later.{{sfn|Dick|1981|pp=75–76}}{{sfn|Kimberling|1981|pp=28–29}} This, however, was before the bloody events of [[Kristallnacht]] in 1938, and their praise from Propaganda Minister [[Joseph Goebbels]].

== Refuge at Bryn Mawr and Princeton, in America ==
[[Mynd:Bryn_Mawr_Sunset.jpg|hægri|thumb|[[Bryn Mawr College]] provided a welcoming home for Noether during the last two&nbsp;years of her life.]]
As dozens of newly unemployed professors began searching for positions outside of Germany, their colleagues in the United States sought to provide assistance and job opportunities for them. [[Albert Einstein]] and [[Hermann Weyl]] were appointed by the [[Institute for Advanced Study]] in [[Princeton, New Jersey|Princeton]], while others worked to find a sponsor required for legal [[immigration]]. Noether was contacted by representatives of two educational institutions: [[Bryn Mawr College]], in the United States, and [[Somerville College]] at the [[University of Oxford]], in England. After a series of negotiations with the [[Rockefeller Foundation]], a grant to Bryn Mawr was approved for Noether and she took a position there, starting in late 1933.{{Sfn|Dick|1981|pp=78–79}}{{Sfn|Kimberling|1981|pp=30–31}}

At Bryn Mawr, Noether met and befriended [[Anna Johnson Pell Wheeler|Anna Wheeler]], who had studied at Göttingen just before Noether arrived there. Another source of support at the college was the Bryn Mawr president, Marion Edwards Park, who enthusiastically invited mathematicians in the area to "see Dr.&nbsp;Noether in action!"{{Sfn|Kimberling|1981|pp=32–33}}{{Sfn|Dick|1981|p=80}} Noether and a small team of students worked quickly through van der Waerden's 1930 book ''Moderne Algebra&nbsp;I'' and parts of [[Erich Hecke]]'s ''Theorie der algebraischen Zahlen'' (''Theory of algebraic numbers'').{{Sfn|Dick|1981|pp=80–81}}

In 1934, Noether began lecturing at the Institute for Advanced Study in Princeton upon the invitation of [[Abraham Flexner]] and [[Oswald Veblen]]. She also worked with and supervised [[Abraham Adrian Albert|Abraham Albert]] and [[Harry Vandiver]].{{Sfn|Dick|1981|pp=81–82}} However, she remarked about [[Princeton University]] that she was not welcome at "the men's university, where nothing female is admitted".{{Sfn|Dick|1981|p=81}}

Her time in the United States was pleasant, surrounded as she was by supportive colleagues and absorbed in her favorite subjects.{{Sfn|Osen|1974|p=151}}{{Sfn|Dick|1981|p=83}} In the summer of 1934 she briefly returned to Germany to see Emil Artin and her brother [[Fritz Noether|Fritz]] before he left for Tomsk. Although many of her former colleagues had been forced out of the universities, she was able to use the library as a "foreign scholar".{{Sfn|Dick|1981|p=82}}{{Sfn|Kimberling|1981|p=34}}

== Death ==
[[Mynd:Bryn_Mawr_College_Cloisters.JPG|hægri|thumb|250x250dp|Noether's remains were placed under the walkway surrounding the cloisters of Bryn Mawr's [[Bryn Mawr College#M. Carey Thomas Library|M. Carey Thomas Library]].]]
In April 1935 doctors discovered a [[tumor]] in Noether's [[Human pelvis|pelvis]]. Worried about complications from surgery, they ordered two days of bed rest first. During the operation they discovered an [[ovarian cyst]] "the size of a large [[cantaloupe]]".{{Sfn|Kimberling|1981|pp=37–38}} Two smaller tumors in her [[uterus]] appeared to be benign and were not removed, to avoid prolonging surgery. For three days she appeared to convalesce normally, and she recovered quickly from a [[circulatory collapse]] on the fourth. On 14 April she fell unconscious, her temperature soared to {{convert|109|°F|°C|sigfig=3}}, and she died. "[I]t is not easy to say what had occurred in Dr. Noether", one of the physicians wrote. "It is possible that there was some form of unusual and virulent infection, which struck the base of the brain where the heat centers are supposed to be located."{{Sfn|Kimberling|1981|pp=37–38}}

A few days after Noether's death her friends and associates at Bryn Mawr held a small memorial service at College President Park's house. Hermann Weyl and [[Richard Brauer]] traveled from Princeton and spoke with Wheeler and Taussky about their departed colleague. In the months that followed, written tributes began to appear around the globe: Albert Einstein<ref>{{cite news|last1=Einstein|first1=Albert|title=The Late Emmy Noether; Professor Einstein Writes in Appreciation of a Fellow-Mathematician.|url=https://query.nytimes.com/gst/abstract.html?res=9807EFDD163CE33ABC4C53DFB366838E629EDE|accessdate=24 March 2015|work=[[The New York Times]]|date=4 May 1935}}</ref> joined van der Waerden, Weyl, and [[Pavel Alexandrov]] in paying their respects. Her body was cremated and the ashes interred under the walkway around the cloisters of the [[Bryn Mawr College#M. Carey Thomas Library|M. Carey Thomas Library]] at Bryn Mawr.{{Sfn|Kimberling|1981|p=39}}

== Contributions to mathematics and physics ==
Noether's work in [[abstract algebra]] and [[topology]] was influential in mathematics, while in physics, [[Noether's theorem]] has consequences for [[theoretical physics]] and [[Dynamical system|dynamical systems]]. She showed an acute propensity for abstract thought, which allowed her to approach problems of mathematics in fresh and original ways.{{Sfn|Osen|1974|pp=148–49}}{{Sfn|Kimberling|1981|pp=11–12}} Her friend and colleague [[Hermann Weyl]] described her scholarly output in three epochs:
{{Quote|Emmy Noether's scientific production fell into three clearly distinct epochs:

(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]]. After moving to Göttingen in 1915, she produced her work for physics, the two [[Noether's theorem|Noether's theorems]].

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.

In these epochs, she was not merely applying ideas and methods of 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 earlier work of [[Richard Dedekind]]. She is also renowned for developing ascending chain conditions, a simple finiteness condition that yielded powerful results in her hands. 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 [[elimination theory]] and the [[Algebraic variety|algebraic varieties]] that had been studied by her father.

=== 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|quintic equations]], as well as on the [[Root of unity|related problem]] of constructing [[Regular polygon|regular polygons]] using [[Compass and straightedge constructions|compass and straightedge]]. Beginning with [[Carl Friedrich Gauss]]'s 1832 proof that [[Prime number|prime numbers]] such as five can be [[Integer factorization|factored]] in [[Gaussian integer|Gaussian integers]],<ref>{{cite journal|first=C.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|permutation groups]] in 1832 (although, because of his death, his papers were published only in 1846, by Liouville), [[William Rowan Hamilton]]'s discovery of [[Quaternion|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]].<ref name="algebra_contributions_paramount">G.E. {{Harvnb|Noether|1987|p=168}}.</ref>

=== Background on abstract algebra and ''begriffliche Mathematik'' (conceptual mathematics) ===
Two of the most basic objects in abstract algebra are groups and rings.

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 [[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 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 [[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. If every non-zero element has a [[multiplicative inverse]] (an element ''x'' such that ''a x''&nbsp;=&nbsp;''x a''&nbsp;=&nbsp;1&nbsp;), the ring is called a ''[[division ring]]''. A ''[[Field (mathematics)|field]]'' is defined as a commutative division ring.

Groups are frequently studied through ''[[Group representation|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 ''[[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.

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]]''. (The word algebra means both a subject within mathematics as well as an 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. 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 [[Word (data type)|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&nbsp;der&nbsp;Waerden recalled that
{{quote|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.

=== Example: Integers as a ring ===
The [[Integer|integers]] 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 ''a'' and ''b'' in the ring, ''a''&nbsp;+&nbsp;''b''&nbsp;=&nbsp;''b''&nbsp;+&nbsp;''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 [[Matrix (mathematics)|matrices]] and [[Quaternion|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&nbsp;×&nbsp;''a''&nbsp;=&nbsp;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 number|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 [[Ideal (ring theory)|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.

=== First epoch (1908–1919): Algebraic invariant theory ===
[[Mynd:Emmy_Noether_-_Table_of_invariants_2.jpg|hægri|thumb|250x250dp|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. As an everyday example, if a rigid yardstick is rotated, the coordinates (''x''<sub>1</sub>, ''y''<sub>1</sub>, ''z''<sub>1</sub>) and (''x''<sub>2</sub>, ''y''<sub>2</sub>, ''z''<sub>2</sub>) of its endpoints change, but its length ''L'' given by the formula  ''L''<sup>2</sup>&nbsp;=&nbsp;Δ''x''<sup>2</sup>&nbsp;+&nbsp;Δ''y''<sup>2</sup>&nbsp;+&nbsp;Δ''z''<sup>2</sup> remains the same. Invariant theory was an active area of research in the later nineteenth century, prompted in part by [[Felix Klein]]'s [[Erlangen program]], according to which different types of [[geometry]] should be characterized by their invariants under transformations, e.g., the [[cross-ratio]] of [[projective geometry]].

An example of an ''invariant'' is the [[discriminant]] ''B''<sup>2</sup>&nbsp;−&nbsp;4 ''A C'' of a binary [[quadratic form]] '''x·'''A '''x'''&nbsp;+&nbsp;'''y·'''B '''x'''&nbsp;+&nbsp;'''y·'''C '''y'''&nbsp;, where '''x''' and '''y''' are [[Euclidean vector|vectors]] and "'''·'''" is the [[dot product]] or "[[inner product]]" for the vectors. A, B, and C are [[Linear operator|linear operators]] on the vectors – typically [[Matrix (mathematics)|matrices]].

The discriminant is called "invariant" because it is not changed by linear substitutions '''''x'''''&nbsp;→&nbsp;''a'' '''''x'''''&nbsp;+&nbsp;''b'' '''''y''''', ''y''&nbsp;→&nbsp;''c'' '''x'''&nbsp;+&nbsp;''d'' '''y''' with determinant ''a'' ''d''&nbsp;−&nbsp;''b'' ''c''&nbsp;=&nbsp;1&nbsp;. These substitutions form the [[special linear group]] ''SL''<sub>2</sub>.<ref group="lower-alpha">There are no invariants under the [[general linear group]] of all invertible linear transformations because these transformations can be multiplication by a scaling factor. To remedy this, classical invariant theory also considered ''relative invariants'', which were forms invariant up to a scale factor.</ref>

One can ask for all polynomials in A, B, and C that are unchanged by the action of ''SL''<sub>2</sub>; these are called the invariants of binary quadratic forms and turn out to be the polynomials in the discriminant.

More generally, one can ask for the invariants of homogeneous polynomials A<sub>0</sub> ''x''<sup>r</sup> ''y''<sup>0</sup>&nbsp;+&nbsp;...&nbsp;+&nbsp;A<sub>r</sub> x<sup>0</sup> ''y''<sup>r</sup> of higher degree, which will be certain polynomials in the coefficients A<sub>0</sub>,&nbsp;...,&nbsp;A<sub>r</sub>, and more generally still, one can ask the similar question for homogeneous polynomials in more than two variables.

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. For example, the discriminant gives a finite basis (with one element) for the invariants of binary quadratic forms.

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=618–21}}{{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}}

=== First epoch (1908–1919): Galois theory ===
[[Galois theory]] concerns transformations of [[Field (mathematics)|number fields]] that [[Permutation|permute]] the roots of an equation. Consider a polynomial equation of a variable ''x'' of [[Degree of a polynomial|degree]] ''n'', in which the coefficients are drawn from some [[ground field]], which might be, for example, the field of [[Real number|real numbers]], [[Rational number|rational numbers]], or the [[Integer|integers]] [[Modular arithmetic|modulo]]&nbsp;7. There may or may not be choices of ''x'', which make this polynomial evaluate to zero. Such choices, if they exist, are called [[Root of a function|roots]]. If the polynomial is ''x''<sup>2</sup>&nbsp;+&nbsp;1 and the field is the real numbers, then the polynomial has no roots, because any choice of ''x'' makes the polynomial greater than or equal to one. If the field is [[Field extension|extended]], however, then the polynomial may gain roots, and if it is extended enough, then it always has a number of roots equal to its degree.

Continuing the previous example, if the field is enlarged to the complex numbers, then the polynomial gains two roots, +''i'' and −''i'', where ''i'' is the [[imaginary unit]], that is, ''i''<sup>&nbsp;2</sup>&nbsp;=&nbsp;−1&nbsp;. More generally, the extension field in which a polynomial can be factored into its roots is known as the [[splitting field]] of the polynomial.

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. (In mathematical jargon, these transformations are called [[Automorphism|automorphisms]].) The Galois group of {{nowrap|''x''<sup>2</sup> + 1}} consists of two elements: The identity transformation, which sends every complex number to itself, and [[complex conjugation]], which sends +''i'' to −''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 ''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|subgroups]] of the Galois group.

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]] ''S''<sub>''n''</sub> acting on the field  ''k''(''x''<sub>1</sub>,&nbsp;...&nbsp;,&nbsp;''x''<sub>''n''</sub>) always is a pure [[transcendental extension]] of the field ''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  ''n''&nbsp;=&nbsp;2, 3, or 4. In 1969, [[R.G. Swan]] found a counter-example to Noether's problem, with  ''n''&nbsp;=&nbsp;47 and ''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>

=== First epoch (1908–1919): 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]], 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 theorem|Noether's first theorem]], which she proved in 1915, but did not publish until 1918.<ref>{{harvnb|Noether|1918b}}</ref> 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:{{quote|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.<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 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 law (physics)|conservation laws]] of [[Momentum|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.<ref name="neeman_1999"/> 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:<blockquote>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.</blockquote>

=== 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 [[Empty set|non-empty]] [[Subset|subsets]] ''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>, etc. of a [[Set (mathematics)|set]] ''S'' is usually said to be ''ascending'', if each is a subset of the next

: <math>A_{1} \subset A_{2} \subset A_{3} \subset \cdots.</math>

Conversely, a sequence of subsets of ''S'' is called ''descending'' if each contains the next subset:

: <math>A_{1} \supset A_{2} \supset A_{3} \supset \cdots.</math>

A chain ''becomes constant after a finite number of steps'' if there is an ''n'' such that <math>A_n = A_m</math> for all ''m''&nbsp;≥&nbsp;''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 (disambiguation)|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 (mathematics)|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 ring|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.

=== Second epoch (1920–1926): 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]].<ref name="Gil133" /> 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.<ref name="Gil133">{{Harvnb|Gilmer|1981|p=133}}.</ref> 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. 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|prime numbers]], and that this decomposition is unique.

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 as the [[Dedekind domain|Dedekind domains]]: integral domains that are Noetherian, 0- or 1[[Krull dimension|-dimensional]], and [[Integrality|integrally closed]] in their quotient fields. This paper also contains what now are called the [[Isomorphism theorem|isomorphism theorems]], which describe some fundamental [[Natural isomorphism|natural isomorphisms]], and some other basic results on Noetherian and [[Artinian module|Artinian modules]].

=== Second epoch (1920–1926): 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, usually by the method of [[Resultant|resultants]].

For illustration, a system of equations often can be written in the form &nbsp; M '''v'''&nbsp;=&nbsp;'''0''' &nbsp; where a matrix (or [[linear transform]]) &nbsp; M &nbsp; (without the variable ''x'') times a vector '''v''' (that only has non-zero powers of ''x'') is equal to the zero vector, '''0'''. Hence, the [[determinant]] of the matrix &nbsp; M &nbsp; must be zero, providing a new equation in which the variable ''x'' has been eliminated.

=== Second epoch (1920–1926): 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 &nbsp; ''G'' &nbsp; 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 &nbsp; |''G''|! &nbsp; (the [[factorial]] of the order &nbsp; |''G''| &nbsp; of the group ''G''). The degrees of generators need not satisfy Noether's bound when the characteristic of the field divides the number &nbsp; |''G''|&nbsp;,{{Sfn|Fleischmann|2000|p=24}} but Noether was not able to determine whether this bound was correct when the characteristic of the field divides &nbsp; |''G''|! &nbsp; but not &nbsp; |''G''|&nbsp;. 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]] ''A'' over a field ''k'' has a set &nbsp; {&nbsp;''x''<sub>1</sub>,&nbsp;...&nbsp;,&nbsp;''x''<sub>''n''</sub>&nbsp;} &nbsp; of [[Algebraic independence|algebraically independent]] elements such that ''A'' is [[Integrality|integral]] over &nbsp; ''k'' [''x''<sub>1</sub>,&nbsp;... ,&nbsp;''x''<sub>''n''</sub>]&nbsp;.

=== Second epoch (1920–1926): Contributions to topology ===
[[Mynd:Mug_and_Torus_morph.gif|hægri|thumb|240x240dp|A continuous deformation ([[homotopy]]) of a coffee cup into a doughnut ([[torus]]) and back]]
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 [[Connected space|connectedness]]. An old joke is that "''a topologist cannot distinguish a donut from a coffee mug''", since they can be continuously deformed into one another.

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 the account of Alexandrov, Noether attended lectures given by [[Heinz Hopf]] and by him 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,
{{quote|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 name="Hirzebruch">[[Friedrich Hirzebruch|Hirzebruch, Friedrich]]. "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=174–75}} 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|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>Hirzebruch, Friedrich, "Emmy Noether and Topology" in {{Harvnb|Teicher|1999|pp=61–63}}.</ref>

[[Mynd:Helmut_Hasse.jpg|hægri|thumb|[[Helmut Hasse]] worked with Noether and others to found the theory of [[Central simple algebra|central simple algebras]].]]

=== Third epoch (1927–1935): Hypercomplex numbers and representation theory ===
Much work on [[Hypercomplex number|hypercomplex numbers]] and [[Group representation|group representations]] was carried out in the nineteenth and early twentieth centuries, but remained disparate. Noether united these results and gave the first general representation theory of groups and algebras.<ref name="noether_1929">{{harvnb|Noether|1929}}.</ref>

Briefly, Noether subsumed the structure theory of [[Associative algebra|associative algebras]] 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|ascending chain conditions]]. This single work by Noether was of fundamental importance for the development of modern algebra.<ref name="vdWaerden_1985">{{harvnb|van der Waerden|1985|p=244}}.</ref>

=== Third epoch (1927–1935): 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|central simple algebras]].{{Sfn|Lam|1981|pp=152–53}}

A paper by Noether, Helmut Hasse, and [[Richard Brauer]] pertains to [[Division algebra|division algebras]],<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 ''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 ''k'' into a finite-dimensional central simple algebra over ''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.

== Assessment, recognition, and memorials ==
{{further|List of things named after Emmy Noether#Other}}
[[Mynd:Emmy-noether-campus_siegen.jpg|hægri|thumb|250x250dp|The Emmy Noether Campus at the [[University of Siegen]] is home to its mathematics and physics departments.]]
Noether's work continues to be relevant for the development of theoretical physics and mathematics and she is consistently ranked as one of the greatest mathematicians of the twentieth century. In his obituary, fellow algebraist [[Bartel Leendert van der Waerden|BL van der Waerden]] says that her mathematical originality was "absolute beyond comparison",{{Sfn|Dick|1981|p=100}} and Hermann Weyl said that Noether "changed the face of [[Abstract algebra|algebra]] by her work".<ref name="weyl_128">{{Harvnb|Dick|1981|p=128}}</ref> During her lifetime and even until today, Noether has been characterized as the greatest woman mathematician in recorded history by mathematicians{{Sfn|Osen|1974|p=152}}{{Sfn|Alexandrov|1981|p=100}}{{Sfn|James|2002|p=321}} such as [[Pavel Alexandrov]],{{Sfn|Dick|1981|p=154}} [[Hermann Weyl]],{{Sfn|Dick|1981|p=152}} and [[Jean Dieudonné]].<ref name="g_noether_p167">{{Harvnb|Noether|1987|p=167}}.</ref>

In a letter to ''[[The New York Times]]'', [[Albert Einstein]] wrote:<ref name="einstein"/>
{{Quote|In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical [[genius]] thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.}}

On 2 January 1935, a few months before her death, mathematician [[Norbert Wiener]] wrote that {{Sfn|Kimberling|1981|p=35}}
{{quote|Miss Noether is ... the greatest woman mathematician who has ever lived; and the greatest woman scientist of any sort now living, and a scholar at least on the plane of [[Marie Curie|Madame Curie]].}}

At an exhibition at the [[1964 World's Fair]] devoted to [[Mathematica: A World of Numbers ... and Beyond|Modern Mathematicians]], Noether was the only woman represented among the notable mathematicians of the modern world.<ref>{{citation|last=Duchin|first=Moon|authorlink=Moon Duchin|url=http://www.math.lsa.umich.edu/~mduchin/UCD/111/readings/genius.pdf|title=The Sexual Politics of Genius|format=PDF|date=December 2004|publisher=University of Chicago|accessdate=23 March 2011|url-status=dead|archiveurl=https://web.archive.org/web/20110718033431/http://www.math.lsa.umich.edu/~mduchin/UCD/111/readings/genius.pdf|archivedate=18 July 2011|df=dmy}} (Noether's birthday).</ref>

Noether has been honored in several memorials,

* The [[Association for Women in Mathematics]] holds a [[Noether Lecture]] to honor women in mathematics every year; in its 2005 pamphlet for the event, the Association characterizes Noether as "one of the great mathematicians of her time, someone who worked and struggled for what she loved and believed in. Her life and work remain a tremendous inspiration".<ref>{{Citation|chapter-url=http://www.awm-math.org/noetherbrochure/Introduction.html|chapter=Introduction|title=Profiles of Women in Mathematics|series=The Emmy Noether Lectures|publisher=[[Association for Women in Mathematics]]|year=2005|accessdate=13 April 2008}}</ref>
* Consistent with her dedication to her students, the [[University of Siegen]] houses its mathematics and physics departments in buildings on ''the Emmy Noether Campus''.<ref>{{Citation|url=http://www.uni-siegen.de/uni/campus/wegweiser/emmy.html|title=Emmy-Noether-Campus|publisher=Universität Siegen|place=[[Germany|DE]]|accessdate=13 April 2008}}</ref>
* The German Research Foundation ([[Deutsche Forschungsgemeinschaft]]) operates the ''Emmy Noether Programme'', providing funding to early-career researchers to rapidly qualify for a leading position in science and research by leading an independent junior research group.<ref>[http://www.dfg.de/en/research_funding/programmes/individual/emmy_noether/index.html "Emmy Noether Programme"]. ''Research Funding''. [[Deutsche Forschungsgemeinschaft]]. n.d. Retrieved on 25 May 2016.</ref>
* A street in her hometown, Erlangen, has been named after Emmy Noether and her father, Max Noether.
* The successor to the secondary school she attended in Erlangen has been renamed as ''the Emmy Noether School''.<ref name="g_noether_p167" />
* A series of high school workshops and competitions are held in her honor in May of each year since 2001, originally hosted by a subsequent woman mathematics [[Privatdozent]] of the [[University of Göttingen]].<ref>Emmy Noether High School Mathematics Days. http://www.math.ttu.edu/~enoether/</ref>
* [[Perimeter Institute for Theoretical Physics]] annually awards Emmy Noether Visiting Fellowships<ref>Emmy Noether Visiting Fellowships http://www.perimeterinstitute.ca/emmy-noether-visiting-fellowships</ref> to outstanding female theoretical physicists. Perimeter Institute is also home to the Emmy Noether Council,<ref>{{cite web|url=https://www.perimeterinstitute.ca/people/emmy-noether-council|title=Emmy Noether Council|website=Perimeter Institute for Theoretical Physics|accessdate=6 March 2018|df=dmy-all}}</ref> a group of volunteers made up of international community, corporate and philanthropic leaders work together to increase the number of women in physics and mathematical physics at Perimeter Institute.
* The Emmy Noether Mathematics Institute in Algebra, Geometry and Function Theory in the Department of Mathematics and Computer Science, [[Bar-Ilan University]], Ramat Gan, Israel was jointly founded in 1992 by the university, the [[German government]] and the [[Minerva Foundation]] with the aim to stimulate research in the above fields and to encourage collaborations with Germany. Its main topics are [[Algebraic Geometry]], [[Group theory]] and [[Complex analysis|Complex Function Theory]]. Its activities includes local research projects, conferences, short-term visitors, post-doc fellowships, and the Emmy Noether lectures (an annual series of distinguished lectures). ENI is a member of ERCOM: "European Research Centers of Mathematics".<ref>The Emmy Noether Mathematics Institute. http://u.cs.biu.ac.il/~eni/</ref>
* In 2013, The European Physical Society established the Emmy Noether Distinction for Women in Physics.<ref>{{Cite web|url=https://www.eps.org/page/distinction_prize_en|title=EPS Emmy Noether Distinction for Women in Physics – European Physical Society (EPS)|website=www.eps.org|access-date=2018-09-14}}</ref> Winners have included Dr [[Catalina Curceanu]], Prof [[Sibylle Günter]] and Prof [[Anne L'Huillier]].

In fiction, Emmy Nutter, the physics professor in "The God Patent" by [[Ransom Stephens]], is based on Emmy Noether.<ref>{{Citation|url=http://ransomstephens.com/the-god-patent.htm|title=The God Patent|first=Ransom|last=Stephens}}</ref>

Farther from home,

* The crater [[Nöther (crater)|Nöther]] on the [[far side of the Moon]] is named after her.
* The minor planet [[7001 Noether]] is named for Emmy Noether.{{Sfn|Schmadel|2003|p=570}}<ref>Blue, Jennifer. [http://planetarynames.wr.usgs.gov/ Gazetteer of Planetary Nomenclature]. [[United States Geological Survey|USGS]]. 25 July 2007. Retrieved on 13 April 2008.</ref>
* [[Google]] put a memorial [[Google doodle|doodle]] created by Google artist [[Sophie Diao]] on Google's homepage in many countries on 23 March 2015 to celebrate Emmy Noether's 133rd birthday.<ref>[https://www.google.com/doodles/emmy-noethers-133rd-birthday Google Doodles: Emmy Noether's 133rd Birthday] 23 March 2015.</ref>

== List of doctoral students ==
{| class="wikitable sortable" id="Noether_doctoral_students"
!Date
!Student name
!Dissertation title and English translation
!University
!Published
|-
|{{nowrap|{{date|1911-12-16|iso}}}}
|Falckenberg, Hans
|Verzweigungen von Lösungen nichtlinearer Differentialgleichungen
{|
|Ramifications of Solutions of Nonlinear Differential Equations<sup>§</sup>
|}
|Erlangen
|Leipzig 1912
|-
|{{nowrap|{{date|1916-03-04|iso}}}}
|Seidelmann, Fritz
|Die Gesamtheit der kubischen und biquadratischen Gleichungen mit Affekt bei beliebigem Rationalitätsbereich
{|
|Complete Set of Cubic and Biquadratic Equations with Affect in an Arbitrary Rationality Domain<sup>§</sup>
|}
|Erlangen
|Erlangen 1916
|-
|{{nowrap|{{date|1925-02-25|iso}}}}
|[[Grete Hermann|Hermann, Grete]]
|Die Frage der endlich vielen Schritte in der Theorie der Polynomideale unter Benutzung nachgelassener Sätze von Kurt Hentzelt
{|
|The Question of the Finite Number of Steps in the Theory of Ideals of Polynomials using Theorems of the Late Kurt Hentzelt<sup>§</sup>
|}
|Göttingen
|Berlin 1926
|-
|{{nowrap|{{date|1926-07-14|iso}}}}
|Grell, Heinrich
|Beziehungen zwischen den Idealen verschiedener Ringe
{|
|Relationships between the Ideals of Various Rings<sup>§</sup>
|}
|Göttingen
|Berlin 1927
|-
|1927
|Doräte, Wilhelm
|Über einem verallgemeinerten Gruppenbegriff
{|
|On a Generalized Conceptions of Groups<sup>§</sup>
|}
|Göttingen
|Berlin 1927
|-
|died before defense
|Hölzer, Rudolf
|Zur Theorie der primären Ringe
{|
|On the Theory of Primary Rings<sup>§</sup>
|}
|Göttingen
|Berlin 1927
|-
|{{nowrap|{{date|1929-06-12|iso}}}}
|[[Werner Weber (mathematician)|Weber, Werner]]
|Idealtheoretische Deutung der Darstellbarkeit beliebiger natürlicher Zahlen durch quadratische Formen
{|
|Ideal-theoretic Interpretation of the Representability of Arbitrary Natural Numbers by Quadratic Forms<sup>§</sup>
|}
|Göttingen
|Berlin 1930
|-
|{{nowrap|{{date|1929-06-26|iso}}}}
|[[Jacob Levitzki|Levitski, Jakob]]
|Über vollständig reduzible Ringe und Unterringe
{|
|On Completely Reducible Rings and Subrings<sup>§</sup>
|}
|Göttingen
|Berlin 1931
|-
|{{nowrap|{{date|1930-06-18|iso}}}}
|[[Max Deuring|Deuring, Max]]
|Zur arithmetischen Theorie der algebraischen Funktionen
{|
|On the Arithmetic Theory of Algebraic Functions<sup>§</sup>
|}
|Göttingen
|Berlin 1932
|-
|{{nowrap|{{date|1931-07-29|iso}}}}
|[[Hans Fitting|Fitting, Hans]]
|Zur Theorie der Automorphismenringe Abelscher Gruppen und ihr Analogon bei nichtkommutativen Gruppen
{|
|On the Theory of Automorphism-Rings of Abelian Groups and Their Analogs in Noncommutative Groups<sup>§</sup>
|}
|Göttingen
|Berlin 1933
|-
|{{nowrap|{{date|1933-07-27|iso}}}}
|[[Ernst Witt|Witt, Ernst]]
|Riemann-Rochscher Satz und Zeta-Funktion im Hyperkomplexen
{|
|The Riemann-Roch Theorem and Zeta Function in Hypercomplex Numbers<sup>§</sup>
|}
|Göttingen
|Berlin 1934
|-
|{{nowrap|{{date|1933-12-06|iso}}}}
|[[Zeng Jiongzhi|Tsen, Chiungtze]]
|Algebren über Funktionenkörpern
{|
|Algebras over Function Fields<sup>§</sup>
|}
|Göttingen
|Göttingen 1934
|-
|1934
|Schilling, Otto
|Über gewisse Beziehungen zwischen der Arithmetik hyperkomplexer Zahlsysteme und algebraischer Zahlkörper
{|
|On Certain Relationships between the Arithmetic of Hypercomplex Number Systems and Algebraic Number Fields<sup>§</sup>
|}
|Marburg
|Braunschweig 1935
|-
|1935
|Stauffer, Ruth
|The construction of a normal basis in a separable extension field
|Bryn Mawr
|Baltimore 1936
|-
|1935
|Vorbeck, Werner
|Nichtgaloissche Zerfällungskörper einfacher Systeme
{|
|Non-Galois [[Splitting field|Splitting Fields]] of Simple Systems<sup>§</sup>
|}
|Göttingen
|
|-
|1936
|Wichmann, Wolfgang
|Anwendungen der p-adischen Theorie im Nichtkommutativen
{|
|Applications of the ''p''-adic Theory in Noncommutative Algebras<sup>§</sup>
|}
|Göttingen
|''Monatshefte für Mathematik und Physik'' (1936) '''44''', 203–24.
|}
-->
== Athugasemdir ==
<!--{{notelist}}
{{reflist|group="lower-alpha"}}-->
<references group="lower-alpha" />

== Tilvísanir ==
{{reflist|20em}}

=== Frekari heimildir ===

* {{cite web|last=Phillips|first=Lee|title=The female mathematician who changed the course of physics—but couldn't get a job|url=https://arstechnica.com/science/2015/05/the-female-mathematician-who-changed-the-course-of-physics-but-couldnt-get-a-job/|website=Ars Technica|publisher=Condé Nast|location=California|date=May 2015}}
* {{cite book|authorlink=Pavel Alexandrov|last=Alexandrov|first=Pavel S.|chapter=In Memory of Emmy Noether|title=Emmy Noether: A Tribute to Her Life and Work|url=https://archive.org/details/emmynoethertribu0000unse|editor1-first=James W|editor1-last=Brewer|editor2-first=Martha K.|editor2-last=Smith|place=New York|publisher=Marcel Dekker|year=1981|isbn=978-0-8247-1550-2|pages=[https://archive.org/details/emmynoethertribu0000unse/page/99 99]–111}}
* {{Citation|last=Byers|first=Nina|authorlink=Nina Byers|chapter=Emmy Noether|title=Out of the Shadows: Contributions of 20th Century Women to Physics|year=2006|editor1-first=Nina|editor1-last=Byers|editor2-first=Gary|editor2-last=Williams|place=Cambridge|publisher=Cambridge University Press|isbn=978-0-521-82197-1}}
* {{Citation|author-last=Dick|author-first=Auguste|authorlink=Auguste Dick|title=Emmy Noether: 1882–1935|place=Boston|publisher=Birkhäuser|year=1981|isbn=978-3-7643-3019-4|translator-first=H.I.|translator-last=Blocher}}
* {{Citation|last1=Fleischmann|first1=Peter|title=The Noether bound in invariant theory of finite groups|doi=10.1006/aima.2000.1952|mr=1800251|year=2000|journal=Advances in Mathematics|volume=156|issue=1|pages=23–32}}
* {{Citation|last1=Fogarty|first1=John|title=On Noether's bound for polynomial invariants of a finite group|url=http://www.ams.org/era/2001-07-02/S1079-6762-01-00088-9/|accessdate=16 June 2008|mr=1826990|year=2001|journal=Electronic Research Announcements of the American Mathematical Society|volume=7|issue=2|pages=5–7|doi=10.1090/S1079-6762-01-00088-9}}
* {{Citation|last=Gilmer|first=Robert|chapter=Commutative Ring Theory|title=Emmy Noether: A Tribute to Her Life and Work|editor1-first=James W.|editor1-last=Brewer|editor2-first=Martha K.|editor2-last=Smith|pages=131–43|place=New York|publisher=Marcel Dekker|year=1981|isbn=978-0-8247-1550-2}}
* {{citation|last1=Gordan|first1=Paul|title=Die simultanen Systeme binärer Formen|language=German|doi=10.1007/BF01444021|year=1870|journal=[[Mathematische Annalen]]|volume=2|issue=2|pages=227–80|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002240513&L=1|url-status=dead|archive-url=https://web.archive.org/web/20140903095153/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002240513&L=1|archive-date=3 September 2014|df=dmy-all}}
* {{Citation|authorlink=William Haboush|first=W.J.|last=Haboush|title=Reductive groups are geometrically reductive|volume=102|year=1975|pages=67–83|doi=10.2307/1970974|issue=1|journal=Annals of Mathematics|jstor=1970974}}
* {{citation|last=Hasse|first=Helmut|authorlink=Helmut Hasse|title=Die Struktur der R.&nbsp;Brauerschen Algebrenklassengruppe über einem algebraischen Zahlkörper|year=1933|language=de|doi=10.1007/BF01448916|journal=Mathematische Annalen|volume=107|pages=731–60|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002276062&L=1|url-status=dead|archive-url=https://web.archive.org/web/20160305072945/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=GDZPPN002276062&L=1|archive-date=5 March 2016|df=dmy-all}}
* {{cite journal|last1=Hilbert|first1=David|author1-link=David Hilbert|title=Ueber die Theorie der algebraischen Formen|language=German|date=December 1890|journal=[[Mathematische Annalen]]|volume=36|issue=4|pages=473–534|doi=10.1007/BF01208503|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0036&DMDID=DMDLOG_0045&L=1|url-status=dead|archive-url=https://web.archive.org/web/20140903165407/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0036&DMDID=DMDLOG_0045&L=1|archive-date=3 September 2014|df=dmy-all}}
* {{cite journal|last=Hilton|first=Peter|year=1988|title=A Brief, Subjective History of Homology and Homotopy Theory in this Century|journal=Mathematics Magazine|volume=60|issue=5|pages=282–91|jstor=2689545}}
* {{cite journal|last1=Hopf|first1=Heinz|author1-link=Heinz Hopf|url=http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1%5BIDDOC%5D=465901|title=Eine Verallgemeinerung der Euler-Poincaréschen Formel|language=de|year=1928|journal=Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse|volume=2|pages=127–36}}
* {{cite book|authorlink=Ioan James|last=James|first=Ioan|year=2002|title=Remarkable Mathematicians from Euler to von Neumann|url=https://archive.org/details/remarkablemathem0000jame|publisher=Cambridge University Press|location=Cambridge|isbn=978-0-521-81777-6}}
* {{Citation|last=Kimberling|first=Clark|chapter=Emmy Noether and Her Influence|pages=3–61|title=Emmy Noether: A tribute to her life and work|editor1-first=James W.|editor1-last=Brewer|editor2-first=Martha K.|editor2-last=Smith|place=New York|publisher=Marcel Dekker|year=1981|isbn=978-0-8247-1550-2}}
* {{Citation|last=Lam|first=Tsit Yuen|chapter=Representation Theory|pages=145–56|title=Emmy Noether: A tribute to her life and work|editor1-first=James W.|editor1-last=Brewer|editor2-first=Martha K.|editor2-last=Smith|place=New York|publisher=Marcel Dekker|year=1981|isbn=978-0-8247-1550-2}}
* {{Citation|author1-link=Leon M. Lederman|last1=Lederman|first1=Leon M.|author2-link=Christopher T. Hill|first2=Christopher T.|last2=Hill|title=Symmetry and the Beautiful Universe|place=Amherst, MA|publisher=Prometheus Books|year=2004|isbn=978-1-59102-242-8}}
* {{Citation|last=Mac Lane|first=Saunders|author-link=Saunders Mac Lane|chapter=Mathematics at the University of Göttingen 1831–1933|title=Emmy Noether: A tribute to her life and work|editor1-first=James W.|editor1-last=Brewer|editor2-first=Martha K.|editor2-last=Smith|pages=65–78|place=New York|publisher=Marcel Dekker|year=1981|isbn=978-0-8247-1550-2}}
* {{Citation|last1=Malle|first1=Gunter|last2=Matzat|first2=Bernd Heinrich|title=Inverse Galois theory|publisher=[[Springer-Verlag]]|location=Berlin, New York|series=Springer Monographs in Mathematics|isbn=978-3-540-62890-3|mr=1711577|year=1999}}
* {{Citation|authorlink=Gottfried E. Noether|last=Noether|first=Gottfried E.|year=1987|title=Women of Mathematics|editor1-last=Grinstein|editor1-first=L.S.|editor2-last=Campbell|editor2-first=P.J.|publisher=Greenwood Press|location=New York|isbn=978-0-313-24849-8}}
* {{citation|last=Noether|first=Max|authorlink=Max Noether|title=Paul Gordan|journal=Mathematische Annalen|volume=75|issue=1|year=1914|pages=1–41|doi=10.1007/BF01564521|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0075&DMDID=DMDLOG_0007&L=1|url-status=dead|archive-url=https://web.archive.org/web/20140904002613/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0075&DMDID=DMDLOG_0007&L=1|archive-date=4 September 2014|df=dmy-all}}
* {{Citation|last=Osen|first=Lynn M.|chapter=Emmy (Amalie) Noether|title=Women in Mathematics|publisher=MIT Press|year=1974|isbn=978-0-262-15014-9|pages=141–52}}
* {{Citation|last=Schmadel|first=Lutz D.|authorlink=Lutz D. Schmadel|title=Dictionary of Minor Planet Names|edition=5th revised and enlarged|place=Berlin|publisher=Springer-Verlag|year=2003|isbn=978-3-540-00238-3}}
* {{cite journal|first=Richard G|last=Swan|title=Invariant rational functions and a problem of Steenrod|journal=Inventiones Mathematicae|year=1969|volume=7|issue=2|pages=148–58|doi=10.1007/BF01389798|bibcode=1969InMat...7..148S}}
* {{cite book|last=Taussky|first=Olga|authorlink=Olga Taussky-Todd|chapter=My Personal Recollections of Emmy Noether|pages=[https://archive.org/details/emmynoethertribu0000unse/page/79 79]–92|title=Emmy Noether: A Tribute to Her Life and Work|url=https://archive.org/details/emmynoethertribu0000unse|editor1-first=James W.|editor1-last=Brewer|editor2-first=Martha K|editor2-last=Smith|place=New York|publisher=Marcel Dekker|year=1981|isbn=978-0-8247-1550-2}}
* {{Citation|authorlink=M. B. W. Tent|author-first=M.B.W.|author-last=Tent|title=Emmy Noether: The Mother of Modern Algebra|year=2008|publisher=[[CRC Press]]}}
* {{citation|authorlink=Bartel Leendert van der Waerden|last=van der Waerden|first=B.L.|title=Nachruf auf Emmy Noether|trans-title=obituary of Emmy Noether|journal=Mathematische Annalen|volume=111|year=1935|language=German|pages=469–74|doi=10.1007/BF01472233|url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0111&DMDID=DMDLOG_0038&L=1|url-status=dead|archiveurl=https://web.archive.org/web/20140903172427/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0111&DMDID=DMDLOG_0038&L=1|archivedate=3 September 2014|df=dmy-all}}. Reprinted in {{harvnb|Dick|1981}}
* {{Citation|last=van der Waerden|first=B.L.|author-mask=3|year=1985|title=A History of Algebra: from al-Khwārizmī to Emmy Noether|publisher=Springer-Verlag|location=Berlin|isbn=978-0-387-13610-3}}
* {{Citation|first=Hermann|last=Weyl|authorlink=Hermann Weyl|title=Emmy Noether|journal=Scripta Mathematica|volume=3|issue=3|pages=201–20|year=1935}} Reprinted as an appendix in {{Harvtxt|Dick|1981}}.
* {{Citation|last1=Weyl|first1=Hermann|title=David Hilbert and his mathematical work|doi=10.1090/S0002-9904-1944-08178-0|mr=0011274|year=1944|journal=[[Bulletin of the American Mathematical Society]]|volume=50|issue=9|pages=612–54}}

{{Wikiquote}}
{{Commonscat}}

{{DEFAULTSORT:Noether, Emmy}}
[[Flokkur:Þýskir stærðfræðingar]]
{{fd|1882|1935}}