return

Celebratio Mathematica

Emmy Noether

Algebra

Amelie Emmy Noether: 1882–1935

In spite of so­cial pre­ju­dices and dis­crim­in­a­tion against wo­men and Jews, Emmy No­eth­er be­came a renowned math­em­atician through her ground­break­ing re­search and dis­cov­er­ies on ab­stract al­gebra, in­vari­ant the­or­ems, ideal the­ory and the the­ory of non­com­mut­at­ive al­geb­ras. The No­eth­er School, No­eth­er Boys and No­eth­eri­an Rings re­flect the im­port­ance and im­pact that her dis­cov­er­ies have in the field of math­em­at­ics and phys­ics.

In an un­usu­al de­par­ture from the tra­di­tion­al way of re­fer­ring to math­em­aticians by their last name, Ame­lie Emmy No­eth­er was warmly known as Emmy in the math­em­at­ic­al com­munity.

Emmy was born on 23 March 1882 in the small uni­versity town of Er­lan­gen in Bav­aria. She had three young­er broth­ers, the old­est named Fritz. Her moth­er, Ida Amalia Kaufmann, a tal­en­ted pi­an­ist, was the daugh­ter of a pros­per­ous mer­chant. Her fath­er, Max No­eth­er, was a dis­tin­guished math­em­atician, a pro­fess­or at the Uni­versity of Er­lan­gen who played an im­port­ant role in the the­ory of al­geb­ra­ic func­tions, as well as in both Emmy’s and Fritz’s pro­fes­sion­al lives. In­deed, Emmy later ex­pan­ded on her fath­er’s in­terest in al­geb­ra­ic func­tions, and ul­ti­mately did some of her most im­port­ant work in that sub­ject. Emmy built on her fath­er’s residue the­or­em: “dur­ing the 1920s she fit­ted this the­or­em in­to her gen­er­al the­ory of ideals in ar­bit­rary rings, help­ing to fur­ther es­tab­lish the ax­io­mat­ic and in­teg­rat­ive tend­en­cies of ab­stract al­geb­ras” [e8]

Emmy’s work in ab­stract al­gebra was ground­break­ing, provid­ing sig­ni­fic­antly new dir­ec­tions in math­em­at­ics. It led her to dis­cov­er the key con­di­tion for rings, the as­cend­ing-chain con­di­tion; such rings are now known as No­eth­eri­an rings. By 1918 she was re­cog­nized for the “ex­treme gen­er­al­ity and ab­stract­ness of ap­proach that [would even­tu­ally be] seen as her most dis­tin­guish­ing char­ac­ter­ist­ic.” [e9].

Emmy’s home life had a strong in­flu­ence on her. Her­mann Weyl, an em­in­ent math­em­atician, friend, and later a col­league of Emmy’s, re­membered Max No­eth­er as a “very in­tel­li­gent, warm-hearted har­mo­ni­ous man of many-sided in­terests and a ster­ling edu­ca­tion” [e14]. Weyl’s ob­ser­va­tion of Emmy was that “[she] her­self was, if I might say so, warm like a loaf of bread. There ir­ra­di­ated from her a broad, com­fort­ing, vi­tal warmth” [e14]. Their home was of­ten filled with math­em­aticians and thought-pro­vok­ing dis­cus­sions, which sparked an in­terest in math­em­at­ics in Emmy. Her warm home life was re­flec­ted by Emmy when she began to teach — she loved her stu­dents; they would come to her apart­ment for tea, take long walks to­geth­er, al­ways talk­ing math­em­at­ics [e11]. Her stu­dents be­came known as the “No­eth­er Boys”.

As a child, Emmy re­ceived the edu­ca­tion typ­ic­al for girls of her time and in­come level, learn­ing house­hold chores, mu­sic and dan­cing, as well as read­ing, writ­ing and arith­met­ic. She did not stand out aca­dem­ic­ally, al­though she dis­played at times an aptitude for lo­gic. She did be­come pro­fi­cient in French and Eng­lish, enough to be­come cer­ti­fied to teach lan­guages in Bav­ari­an girls’ schools. But Emmy’s in­terest lay in math­em­at­ics, and she chose to con­tin­ue her edu­ca­tion at the Uni­versity of Er­lan­gen, where her broth­er Fritz was already a stu­dent. Wo­men could at­tend lec­tures only un­of­fi­cially and only with the ex­press per­mis­sion of the lec­turer. Nev­er­the­less, she passed her gradu­ation ex­am­in­a­tion in Ju­ly 1903, and the fol­low­ing semester she went to the Uni­versity of Göt­tin­gen. When the re­stric­tions against fe­male en­roll­ment were dropped, Emmy trans­ferred back to Er­lan­gen.

A fre­quent vis­it­or to the No­eth­er home was Paul Gordan, a fam­ily friend and pro­fess­or at Er­lan­gen. Un­der Gordan’s su­per­vi­sion she wrote her dis­ser­ta­tion [e1], ex­plain­ing “new prop­er­ties she dis­covered of tern­ary bi­quad­rat­ic forms, al­geb­ra­ic op­er­at­ors re­lated to poly­no­mi­als with three vari­ables in which the ex­po­nents in every item add up to four” [e15]. (Emmy dis­missed her thes­is as a “jumble of for­mu­las” [e8].) She suc­cess­fully de­fen­ded her thes­is, and in Decem­ber 1907, at the age of 26, re­ceived her doc­tor­ate in math­em­at­ics summa cum laude. She was one of the first Ger­man wo­men to earn a Ph.D. in any sub­ject [e15].

Hav­ing over­come re­stric­tions on wo­men’s edu­ca­tion, Emmy was now faced with the re­stric­tions placed on wo­men’s em­ploy­ment, par­tic­u­larly in the aca­dem­ic sphere. The nat­ur­al next step for a re­cip­i­ent of a doc­tor­al de­gree would have been the Ha­bil­it­a­tion (an in­de­pend­ent thes­is at a level above the Ph.D.), the fi­nal step ne­ces­sary for a can­did­ate to be­come qual­i­fied to lec­ture at Ger­man uni­versit­ies. As a wo­man, Emmy was barred from this path. In­stead, she taught at the Math­em­at­ic­al In­sti­tute of Er­lan­gen for sev­er­al years without pay or a title, some­times sub­sti­tut­ing for her fath­er when he was too ill to teach. She also began to con­duct her own re­search; al­though an un­of­fi­cial fac­ulty mem­ber, two stu­dents re­ceived their doc­tor­ates un­der her dir­ec­tion [e15]. Gordan’s suc­cessor, al­geb­ra­ist Ernst Fisc­her, was par­tic­u­larly in­flu­en­tial in trans­ition­ing Emmy’s fo­cus from Gordan’s form­al­ist ap­proach to her now-fam­ous ax­io­mat­ic ap­proach. Weyl’s ob­ser­va­tion is in­ter­est­ing [e14]:

It is queer enough that a form­al­ist like Gordan was the math­em­atician from whom her math­em­at­ic­al or­bit set out; a great­er con­trast is hardly ima­gin­able between her first pa­per, the dis­ser­ta­tion, and her works of ma­tur­ity; for the former is an ex­treme ex­ample of form­al com­pu­ta­tions and the lat­ter con­sti­tute an ex­treme and gran­di­ose ex­ample of con­cep­tu­al ax­io­mat­ic think­ing in math­em­at­ics.

Two pa­pers she wrote in 1913, “Ra­tion­al func­tion fields” [e2], and an ex­ten­ded pa­per on “Fields and sys­tems of ra­tion­al func­tions” [e3], along with her doc­tor­al dis­ser­ta­tion, “es­tab­lished Emmy’s repu­ta­tion in the field of in­vari­ant the­ory, which is where re­search­ers study prop­er­ties that re­main fixed when an ob­ject is sub­jec­ted to modi­fy­ing trans­form­a­tions” [e15].

In 1915, Hil­bert and Fe­lix Klein in­vited Emmy to re­turn to the Uni­versity of Göt­tin­gen to work with them on the re­search they were con­duct­ing on ap­plic­a­tions of in­vari­ant the­ory. Ein­stein, a phys­ics pro­fess­or at the Uni­versity of Ber­lin, had already for­mu­lated his the­ory on re­lativ­ity, and Klein and Hil­bert were “try­ing to de­term­ine the field equa­tions for gen­er­al re­lativ­ity to de­scribe the prop­er­ties of a grav­it­a­tion­al field sur­round­ing a giv­en mass” [e15].

Dur­ing the next four years, Emmy wrote nine pa­pers on vari­ous as­pects of in­vari­ant the­ory. One on the Galois group of poly­no­mi­al equa­tions “rep­res­en­ted the most sig­ni­fic­ant con­tri­bu­tion to the solu­tion of this clas­sic prob­lem” [e15].

Emmy’s 1918 pa­per on “In­vari­ant vari­ation­al prob­lems” [e4] is her most widely known work. It con­tains No­eth­er’s The­or­em, which ex­plains the re­la­tion­ship between phys­ic­al sym­metry and con­ser­va­tion laws, and is con­sidered to be of great im­port­ance in the de­vel­op­ment of mod­ern phys­ics.

It shows how sym­met­ries in a phys­ic­al sys­tem lead to con­served quant­it­ies. In the pa­per, Emmy con­siders phys­ic­al sys­tems based on an ac­tion prin­ciple. That is, the laws of mo­tion come from min­im­iz­ing some ac­tion func­tion­al. (This is the case for most mech­an­ic­al sys­tems.) She then adds the as­sump­tion that there is a con­tinu­ous group of sym­met­ries of the phys­ic­al sys­tem, i.e., the ac­tion of a Lie group on the con­fig­ur­a­tion space, leav­ing the ac­tion func­tion­al un­changed. From this, she de­rives ex­pli­cit quant­it­ies that are con­stant along the phys­ic­al tra­ject­or­ies. The num­ber of such in­de­pend­ent quant­it­ies equals the di­men­sion of the Lie group.

An ex­ample is a mech­an­ic­al sys­tem of \( n \) particles whose in­ter­ac­tions are un­changed by a sim­ul­tan­eous trans­la­tion in 3-space of all of the particles, or by the ro­ta­tion around a point of all of the particles. The con­served quant­it­ies com­ing from the trans­la­tion sym­met­ries are the total mo­menta. The con­served quant­it­ies com­ing from the ro­ta­tions are the total an­gu­lar mo­menta.

Emmy’s pa­per is framed in the ab­stract set­ting of vari­ation­al cal­cu­lus. She does not dis­cuss phys­ic­al ex­amples, al­though her work was in­spired by the on­go­ing ef­forts of Hil­bert to de­rive the laws of gen­er­al re­lativ­ity from an ac­tion prin­ciple.

However, due to the ob­jec­tions of oth­er fac­ulty mem­bers, Emmy’s role was again re­stric­ted. The pro­fess­ors felt that it would be hu­mi­li­at­ing to re­quire male stu­dents to be taught by a fe­male pro­fess­or. She was still not paid for her work, nor did she have an of­fi­cial title. She gave lec­tures that were ad­vert­ised un­der Hil­bert’s name (“with the as­sist­ance of Dr. E. No­eth­er”). Hil­bert was so frus­trated and angry over be­ing un­able to se­cure a paid po­s­i­tion for Emmy at Göt­tin­gen that he de­clared at a fac­ulty meet­ing, “I do not see that the sex of the can­did­ate is an ar­gu­ment against her ad­mis­sion as a Privat­dozent. After all, we are a uni­versity and not a bathing es­tab­lish­ment” {mr:2118372}.

In 1919, after the end of the war, con­di­tions changed and Emmy was giv­en a title of “Pro­fess­or”; she could now lec­ture un­der her own name, but with no salary. In 1923, thanks to her close math­em­at­ic­al col­leagues, she re­ceived an of­fi­cial com­mis­sion to teach, that would pay her a small salary. It was dur­ing this time that the “No­eth­er Boys” be­came a phe­nomen­on. This was a group of ten male stu­dents who ad­mired her and her math­em­at­ic­al bril­liance, and who re­ceived their doc­tor­al de­grees in math­em­at­ics un­der her su­per­vi­sion.

In 1920 she co-au­thored (with Schmeidler) a pa­per on dif­fer­en­tial op­er­at­ors [e5] that changed the face of al­gebra. Its im­me­di­ate in­flu­ence was minor, but look­ing back this is where her con­cep­tu­al ax­io­mat­ic way of think­ing truly sur­faces.

Emmy’s pa­pers from 1920 to 1926 em­phas­ized ab­stract prop­er­ties of groups, rings, fields, ideals, and mod­ules, rather than the spe­cif­ic ob­jects them­selves. Her mo­nu­ment­al 1921 pa­per “The the­ory of ideals in ring do­mains” [e6] has be­come the basis for com­mut­at­ive ring the­ory. She in­tro­duced what are now com­monly re­ferred to as No­eth­eri­an rings and No­eth­eri­an ideals.

Emmy’s re­search on ab­stract al­gebra was a mag­net for many stu­dents and pro­fess­ors. She and her group be­came known as the No­eth­er School, and as a res­ult the Math­em­at­ic­al In­sti­tute at Göt­tin­gen be­came the world’s most re­spec­ted and in­flu­en­tial cen­ter for math­em­at­ic­al re­search.

Over a peri­od of time, be­gin­ning with the winter of 1925–26 when she vis­ited Hol­land and talked with L. E. J. Brouwer and P. S. Aleksandrov, and later in courses and talks with Heinz Hopf, Emmy in­flu­enced to­po­lo­gists to study ho­mo­logy groups as quo­tients of the group of cycles by the sub­group of bound­ar­ies, rather than simply com­put­ing Betti num­bers [e12]. Aleksandrov and Hopf, in the pre­face to their 1935 book To­po­lo­gie, I [e7], wrote:

Emmy No­eth­er’s over­all math­em­at­ic­al in­sight was not lim­ited to al­gebra, her par­tic­u­lar spe­cialty, but ex­er­ted an en­liven­ing in­flu­ence on any­one who had math­em­at­ic­al con­tact with her. […] [W]e fol­low the trend to­wards a thor­ough “al­geb­ra­iciz­a­tion” of to­po­logy, based on group the­ory, which goes dir­ectly back to Emmy No­eth­er. Today, this de­vel­op­ment ap­pears self-evid­ent; eight years ago it was not. It took Emmy No­eth­er’s en­ergy and tem­pera­ment to make al­geb­ra­ic think­ing part of the to­po­lo­gist’s rep­er­toire, and to al­low al­geb­ra­ic prob­lems and meth­ods to play the role they do in to­po­logy today.

Dur­ing 1927–1935, her ab­stract fo­cus was re­dir­ec­ted to the study of non­com­mut­at­ive al­geb­ras. Dur­ing this peri­od, she wrote 13 pa­pers in this area, one of which Her­mann Weyl re­garded as “a high wa­ter­mark in the de­vel­op­ment of al­gebra” [e15].

In 1932, two events af­firmed the re­spect for Emmy with­in the math­em­at­ics com­munity: she was awar­ded the Al­fred Ack­er­mann-Taub­n­er Me­mori­al Award for the ad­vance­ment of math­em­at­ic­al sci­ences, and in Septem­ber was in­vited to de­liv­er one of the main ad­dresses at the In­ter­na­tion­al Math­em­at­ic­al Con­gress (ICM) in Switzer­land.

When the Nazis came to power in Ger­many, Emmy was again barred from teach­ing at the uni­versity. She had three strikes against her: she was Jew­ish, an in­tel­lec­tu­al wo­man, and a philo­soph­ic­al act­iv­ist. “The ser­i­ous struggles that shook Ger­many dur­ing these years helped shape Emmy’s philo­sophy as a pa­ci­fist, an at­ti­tude she held very strongly for the rest of her life” [e8]. In late 1933 and with a grant from the Rock­e­feller Found­a­tion, Emmy went to Bryn Mawr Col­lege as a pro­fess­or. She also began lec­tur­ing at the In­sti­tute for Ad­vanced Study in Prin­ceton. (Her broth­er Fritz went to the Re­search In­sti­tute for Math­em­at­ics and Mech­an­ics in Tomsk, Siber­ia.)

In early 1935, however, Emmy had to have an op­er­a­tion to re­move a tu­mor in her pel­vis. Doc­tors dur­ing the op­er­a­tion found a large ovari­an cyst, which they re­moved, and two smal­ler tu­mors ap­peared to be be­nign. After four days of ap­par­ently nor­mal post-op­er­at­ive re­cu­per­a­tion, Emmy died un­ex­pec­tedly. She was only 53 years old and at the peak of her pro­duct­ive power and tech­nic­al skill.

Her old friend who dearly loved her, Her­mann Weyl, de­livered the me­mori­al ad­dress at Bryn Mawr Col­lege on April 26, 1935. Here’s an ex­cerpt:

It was only too easy for those who met her for the first time, or had no feel­ing for her cre­at­ive power, to con­sider her queer and to make fun at her ex­pense. She was heavy of build and loud of voice, and it was of­ten not easy for one to get the floor in com­pet­i­tion with her. She preached migh­tily, and not as the scribes. She was a rough and simple soul, but her heart was in the right place. Her frank­ness was nev­er of­fens­ive in the least de­gree. In every­day life she was most un­as­sum­ing and ut­terly un­selfish; she had a kind and friendly nature. Nev­er­the­less she en­joyed the re­cog­ni­tion paid her; she could an­swer with a bash­ful smile like a young girl to whom one had whispered a com­pli­ment. No one could con­tend that the Graces had stood by her cradle; but if we in Göt­tin­gen of­ten chaff­ingly re­ferred to her as “der No­eth­er” (with the mas­cu­line art­icle), it was also done with a re­spect­ful re­cog­ni­tion of her power as a cre­at­ive thinker who seemed to have broken through the bar­ri­er of sex. She pos­sessed a rare hu­mor and sense of so­ci­ab­il­ity; a tea in her apart­ments could be most pleas­ur­able. But she was a one-sided who was thrown out of bal­ance by the over­weight of her math­em­at­ic­al tal­ent. […] The memory of her work in sci­ence and of her per­son­al­ity among her fel­lows will not soon pass away. She was a great math­em­atician, the greatest, I firmly be­lieve, that her sex has ever pro­duced and a great wo­man.

A more com­plete ab­stract can be read on this site

Her friend and col­league, Al­bert Ein­stein, said

In the judg­ment of the most com­pet­ent liv­ing math­em­aticians, Fraulein 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.

The com­plete let­ter fol­lows.

Einstein’s letter to the New York Times

To the Ed­it­or of The New York Times:

The ef­forts of most hu­man be­ings are con­sumed in the struggle for their daily bread, but most of those who are, either through for­tune or some spe­cial gift, re­lieved of this struggle are largely ab­sorbed in fur­ther im­prov­ing their worldly lot. Be­neath the ef­fort dir­ec­ted to­ward the ac­cu­mu­la­tion of worldly goods lies all to fre­quently the il­lu­sion that this is the most sub­stan­tial and de­sir­able end to be achieved; but there is, for­tu­nately, a minor­ity com­posed of those who re­cog­nize early in their lives that the most beau­ti­ful and sat­is­fy­ing ex­per­i­ences open to hu­man­kind are not de­rived from the out­side, but are bound up with the de­vel­op­ment of the in­di­vidu­al’s own feel­ing, think­ing and act­ing. The genu­ine artists, in­vest­ig­at­ors and thinkers have al­ways been per­sons of this kind. However in­con­spicu­ously the life of these in­di­vidu­als runs its course, none the less the fruits of their en­deavors are the most valu­able con­tri­bu­tions which one gen­er­a­tion can make to its suc­cessors.

With­in the past few days a dis­tin­guished math­em­atician, Pro­fess­or Emmy No­eth­er, formerly con­nec­ted with the Uni­versity of Göt­tin­gen and for the past two years at Bryn Mawr Col­lege, died in her fifty-third year. In the judg­ment of the most com­pet­ent liv­ing math­em­aticians, Fraulein 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. Pure math­em­at­ics is, in its way, the po­etry of lo­gic­al ideas. One seeks the most gen­er­al ideas of op­er­a­tion which will bring to­geth­er in simple, lo­gic­al and uni­fied form the largest pos­sible circle of form­al re­la­tion­ships. In this ef­fort to­ward lo­gic­al beauty spir­itu­al for­mu­lae are dis­covered ne­ces­sary for the deep­er pen­et­ra­tion in­to the laws of nature.

Born in a Jew­ish fam­ily dis­tin­guished for the love of learn­ing, Emmy No­eth­er, who, in spite of the ef­forts of the great Göt­tin­gen, Hil­bert, nev­er reached the aca­dem­ic stand­ing due her in her own coun­try, none the less sur­roun­ded her­self with a group of stu­dents and in­vest­ig­at­ors at Göt­tin­gen, who have already be­come dis­tin­guished as teach­ers and in­vest­ig­at­ors. Her un­selfish, sig­ni­fic­ant work over a peri­od of many years was re­war­ded by new rulers of Ger­many with a dis­missal, which cost her the means of main­tain­ing her simple life and the op­por­tun­ity to carry on her math­em­at­ic­al stud­ies. Farsighted friends of sci­ence in this coun­try were for­tu­nately able to make such ar­range­ments at Bryn Mawr Col­lege and at Prin­ceton that she found in Amer­ica up to the day of her death not only col­leagues who es­teemed her friend­ship but grate­ful pu­pils whose en­thu­si­asm made her last years the hap­pi­est and per­haps the most fruit­ful of her en­tire ca­reer.

Al­bert Ein­stein,
Prin­ceton Uni­versity, May 1, 1935.

Sev­enty six years after the un­timely death of Emmy No­eth­er, it is dif­fi­cult to say much new, giv­en the wealth of ma­ter­i­al writ­ten about her. The most com­plete work is of course her col­lec­ted works [e10], with its fine in­tro­duc­tion by Nath­an Jac­ob­son. Also, the book [e9] ed­ited by Brew­er and Smith has many in­ter­est­ing art­icles writ­ten on the oc­ca­sion of her one-hun­dredth birth­day. A 1996 con­fer­ence pro­ceed­ings from Bar-Il­an Uni­versity [e13] has fine art­icles on Emmy’s work and in­flu­ence. For those in­ter­ested in Emmy’s in­flu­ence on to­po­logy, [e16] is a good place to start.