埃米·诺特Emmy Noether

埃米·诺特的父亲马克斯·诺特是一位杰出的数学家，也是埃尔朗根的一位教授，但他出身于一个五金批发商家庭。她的母亲是Ida Amalia Kaufmann（1852-1915），来自科隆一个富裕家庭。埃米·诺特的父母都是犹太裔，读者可能会对此感到惊讶，因为埃米·诺特并不是一个犹太名字。因此，我们应该解释这是怎么发生的，同时提供一些关于埃米·诺特祖先的信息。马克斯·诺特的祖父是Elias Samuel，他在布鲁赫萨尔创办了一家商行。Elias有九个孩子，其中一个儿子叫Hertz 皮埃尔·萨米埃尔。1809年，巴登州颁布了《宽容法令》，要求犹太人采用日耳曼名字。Elias Samuel选择了Nöther这个姓氏，成为Elias Nöther，但也更改了孩子们的名字，给Hertz取名为雅各布·赫尔曼。十八岁时，赫尔曼 Nöther离开了家乡布鲁赫萨尔，在曼海姆大学学习神学。然后在1837年，他和他的兄弟Joseph一起创办了一家铁五金批发商行。赫尔曼 Nöther和他的妻子Amalia有五个孩子，第三个是Max。比Max大的两个孩子是Sarah（生于1839年11月6日）和Emil。此时值得注意的是，Nöther铁批发商行作为家族企业整整持续了一百年，直到1937年纳粹将犹太家庭从他们自己的企业中清除出去。此时还有一点需要说明。尽管Max的祖父选择了Nöther作为家族姓氏，但Max和他的家人一直使用埃米·诺特这一形式（除了在Max的结婚证书上出现了Nöther这一形式）。

埃米·诺特是她父母四个孩子中最大的，另外三个孩子都是男孩。Alfred Noether（1883-1918）学习化学，并于1909年在埃尔朗根获得博士学位。然而，他的职业生涯很短，因为九年后他就去世了。弗里茨·诺特（1884-1941）成为一名应用数学家。然而，作为犹太人，他无法工作，并于1937年离开德国。他被任命为苏联托木斯克大学的教授，但被指控从事反苏活动，被判处死刑并被枪决。1988年，苏联最高法院判定他无罪。Gustav Robert Noether（1889-1928）一生健康状况不佳。他有智力障碍，一生大部分时间都在机构中度过，并英年早逝。埃米·诺特上的第一所学校位于Fahrstrasse。Auguste Dick写道[5]：-

埃米·诺特小时候并不显得特别出众。在Fahrstrasse的校园里与同龄人玩耍时，她可能并不特别引人注目——一个近视、相貌平平的小女孩，但并非没有魅力。她的老师和同学都知道埃米·诺特是一个聪明、友好、可爱的孩子。她有点口齿不清，是少数参加犹太宗教课程的人之一。

小学毕业后，埃米·诺特从1889年到1897年在埃尔朗根Friedrichstrasse的Städtische Höhere Töchter Schule就读。她出生在Hauptstrasse 23号的家中，并一直住在那里，直到高中期间，1892年，全家搬到Nürnberger Strasse 32号一个更大的公寓。在高中，她学习德语、英语、法语、算术，并接受钢琴课。她喜欢跳舞，并期待与父亲大学同事的孩子们一起参加聚会。在这个阶段，她的目标是成为一名语言教师，在进一步学习英语和法语后，她参加了巴伐利亚州的考试，并于1900年成为巴伐利亚女子学校的认证英语和法语教师。她在考试中获得了“非常优秀”的成绩，最弱的部分是课堂教学。

埃米·诺特从未成为语言教师。相反，她决定走当时女性少有人走的路，在大学学习数学。女性被允许在德国大学非正式地学习，每位教授必须允许其课程。埃米·诺特在1900至1902年间获得许可在埃尔朗根大学旁听课程。她是埃尔朗根仅有的两名女旁听生之一，除了数学课程外，她还继续对语言的兴趣，由罗马研究教授和一位历史学家授课。同时，她准备参加允许学生进入任何大学的考试。1903年7月14日在纽伦堡参加并通过了这项入学考试后，她去了哥廷根大学。在1903-04年间，她听了卡尔·史瓦西、奥托·布卢门塔尔、大卫·希尔伯特、菲利克斯·克莱因和赫尔曼·闵可夫斯基的讲座。她再次不被允许成为正式注册的学生，只被允许旁听讲座。在哥廷根一个学期后，她回到了埃尔朗根。

这时规则改变了，女学生被允许与男生平等地注册入学。1904年10月24日，埃米·诺特在埃尔朗根注册入学，此时她只学习数学。1907年，她在保罗·哥尔丹指导下工作后获得了博士学位。口试于12月13日星期五进行，她被授予“最优等”学位。大卫·希尔伯特1888年的基定理给出了$n$个变量中不变量的有限性的存在性结果。然而，保罗·哥尔丹采取了构造性的方法，并着眼于构造性方法来达到相同的结果。埃米·诺特的博士论文遵循了保罗·哥尔丹的这种构造性方法，并列出了331个协变形式的系统。Colin McLarty写道[39]：-

……她1908年与保罗·哥尔丹合作的学位论文进行了一项巨大的计算，这项计算在四十年前曾难倒保罗·哥尔丹，而埃米·诺特也无法完成。据我所知，没有人曾经完成它，甚至没有人检查过她所做到的程度。这在当时是过时的，是埃尔朗根令人愉快的与世隔绝的见证，并且没有利用保罗·哥尔丹自己在大卫·希尔伯特思想基础上所做的工作。

完成博士学位后，正常的晋升到学术职位的途径本应是教授资格论文（Habilitation）。然而这条道路对女性不开放，所以埃米·诺特留在埃尔朗根，帮助她的父亲，特别是因为他自己的残疾，他感激女儿的帮助。埃米·诺特也从事自己的研究，特别是她受到Ernst Fischer的影响，后者在1911年退休时接替保罗·哥尔丹担任数学讲席。埃米·诺特写到Fischer的影响：-

首先，我感谢E Fischer先生，我从他那里得到了从算术观点研究抽象代数的决定性推动力，这成为我后来所有工作的主导思想。

Fischer的影响将埃米·诺特引向大卫·希尔伯特对该学科的抽象方法，而远离保罗·哥尔丹的构造性方法。现在这对她作为数学家的成长非常重要，因为保罗·哥尔丹尽管取得了非凡的成就，但也有其局限性。埃米·诺特的父亲马克斯·诺特谈到保罗·哥尔丹时说（见[3]）：-

保罗·哥尔丹从未能够公正地对待基本概念的发展；即使在他的讲座中，他也完全避免了所有概念性质的基本定义，甚至极限的定义。

埃米·诺特的声誉随着她的著作问世而迅速增长。1908年，她当选为Circolo Matematico di Palermo会士，随后在1909年受邀成为Deutsche Mathematiker-Vereinigung成员，同年受邀在萨尔茨堡的学会年会上发表演讲。她作了题为Zur Invariantentheorie der Formen von n Variabeln Ⓣ（论n变量形式的不变量理论）的讲座。1913年，她在维也纳再次向Deutsche Mathematiker-Vereinigung的会议发表演讲。她这次的演讲题目是Über rationale Funktionenkörper Ⓣ（论有理函数域）。在维也纳期间，她拜访了弗朗茨·梅滕斯并与他讨论数学。Merten的一个孙子记得埃米·诺特的来访（见[5]）：-

……尽管是女性，[她]在我看来却像一位来自乡村教区的天主教神父——身穿一件黑色、几乎及踝且相当不起眼的外套，短发上戴着一顶男式帽子……还斜挎着一个像帝国时期铁路列车员那样的肩包，她是个相当古怪的人物。

在埃尔朗根这些年里，她指导了两名博士生，但两人名义上均由她的父亲正式指导。他们是Hans Falckenberg（1911年获博士学位）和Fritz Seidelmann（1916年获博士学位）。

关于这些以及埃米·诺特的其他博士生信息，见THIS LINK。

1915年，大卫·希尔伯特和菲利克斯·克莱因邀请埃米·诺特返回哥廷根。原因是大卫·希尔伯特正在研究物理学，特别是与阿尔伯特·爱因斯坦相近的相对论思想。他认定自己需要一位不变量理论专家的帮助，在与菲利克斯·克莱因商议后，他们发出了邀请。巴特尔·伦德特·范德瓦尔登写道[68]：-

她来了，并立即解决了两个重要问题。第一：如何获得黎曼空间中任意向量场或张量场的所有微分协变量？……埃米·诺特研究的第二个问题来自狭义相对论。她证明了：洛伦兹群的每一个无穷小变换都对应一个守恒定理。

这一理论物理学的成果有时被称为埃米·诺特定理，它证明了物理学中的对称性与守恒原理之间的关系。这一相对论中的基本成果受到阿尔伯特·爱因斯坦在给大卫·希尔伯特的信中的赞扬，他在信中提到了埃米·诺特敏锐的数学思维。当然，她是在第一次世界大战期间来到哥廷根的。那是一个极其艰难的时期，这些年她生活贫困，政治上她成了激进的社会主义者。然而，就数学而言，这些年对她来说异常丰硕。赫尔曼·外尔在[69]中写到了埃米·诺特的政治观点：-

在1918年革命后的动荡时期，她并未远离政治激情，或多或少地站在社会民主党一边；虽然没有实际参与党派生活，但她积极参与了当时政治和社会问题的讨论。……在后来的岁月里，埃米·诺特没有参与政治事务。然而，她始终是一个坚定的和平主义者，她认为这一立场非常重要和严肃。

大卫·希尔伯特和菲利克斯·克莱因说服她留在哥廷根，同时他们为她争取正式教职而斗争。在与大学当局的长期斗争中，允许埃米·诺特获得任教资格的请求遭遇了许多挫折，直到1919年才获得许可，并被授予Privatdozent的职位。在此期间，大卫·希尔伯特允许埃米·诺特以自己的名义宣传她的课程来进行授课。例如，1916-17年冬季学期开设的一门课程在目录中显示为：-

数学物理讨论班：大卫·希尔伯特教授，由E 埃米·诺特博士协助，每周一4-6点，免学费。

在哥廷根，1919年之后，埃米·诺特从不变量理论转向理想理论研究，产生了一种抽象理论，帮助将环论发展为一个主要的数学主题。Idealtheorie in Ringbereichen Ⓣ（环的理想理论）（1921年）在现代代数的发展中具有根本重要性。在这篇论文中，她给出了在满足升链条件的任何交换环中，理想分解为初等理想的交。伊曼纽·拉斯克（后来成为世界象棋冠军）已经证明了域上多项式环的这一结果。埃米·诺特于1924年发表了Abstrakter Aufbau der Idealtheorie in algebraischen Zahlkörpern Ⓣ（代数数域中理想理论的抽象组合）。在这篇论文中，她给出了环上的五个条件，使她能够推导出在这样的交换环中，每个理想都是素理想的唯一乘积。

在1924年同年，巴特尔·伦德特·范德瓦尔登来到哥廷根，与埃米·诺特一起学习了一年。回到阿姆斯特丹后，巴特尔·伦德特·范德瓦尔登写了他的书Moderne AlgebraⓉ（现代代数），共两卷。第二卷的主要部分由埃米·诺特的工作组成。从1927年起，埃米·诺特与哈塞和理查德·布饶尔合作研究非交换代数。他们写了一篇漂亮的联合论文Beweis eines Hauptsatzes in der Theorie der AlgebrenⓉ（代数理论中一个基本定理的证据），于1932年发表。除了教学和研究，埃米·诺特还帮助编辑Mathematische Annalen。她的许多工作出现在同事和学生撰写的论文中，而不是以她自己的名义。

对她杰出数学贡献的进一步认可来自于受邀在1928年9月博洛尼亚和1932年9月苏黎世的国际数学家大会上发表演讲。她在1932年大会上的演讲题为Hyperkomplexe Systeme in ihren Beziehungen zur kommutativen Algebra und zur Zahlentheorie Ⓣ（超复数系统与交换代数和数论的关系）。1932年，她还与埃米尔·阿廷共同获得了Alfred Ackermann-Teubner数学知识进步纪念奖。1933年4月，当纳粹因她是犹太人而将她从哥廷根大学解雇时，她的数学成就毫无用处。她没有收到养老金或任何其他形式的补偿，但她认为自己比其他人更幸运。她于1933年5月10日写信给哈塞（例如见[5]）：-

非常感谢你充满同情的来信！不过我必须说，这件事对我来说远没有对许多其他人那么可怕。至少我有一小笔遗产（反正我从未有资格领取养老金），这让我可以暂时坐下来看看。

赫尔曼·外尔在她葬礼上的讲话中谈到了埃米·诺特对周围发生的可怕事件的反应：-

你不相信邪恶，确实你从未想到它会在人类事务中起作用。这一点在我们一起在哥廷根度过的最后一个夏天，即1933年那个风暴般的夏天，再清楚不过地向我表明了。在我们周围各派系进行的可怕斗争、毁灭和剧变中，在仇恨与暴力、恐惧与绝望与沮丧的海洋中——你走自己的路，像以前一样勤勉地思考数学的挑战。当你被禁止使用研究所的讲堂时，你把学生聚集在自己家里。即使是那些穿褐衫的人也被欢迎；你从未有一秒钟怀疑他们的正直。不顾自己的命运，敞开心扉，无所畏惧，总是和解，你走自己的路。我们许多人相信一种敌意已被释放，其中不可能有宽恕；但你却未受其影响。

关于赫尔曼·外尔演讲的一个版本，见THIS LINK。

她接受了美国布林莫尔学院为期一年的访问教授职位，并于1933年10月乘Bremen号船前往美国就任。她本希望推迟接受邀请，因为她想去英国的牛津，但很快清楚她必须迅速离开。在布林莫尔，她受到数学系主任安娜·约翰逊·佩尔·惠勒的热烈欢迎。埃米·诺特在1933-34年冬季学期为三名学生和一名教员举办了一个讨论班。他们研读了巴特尔·伦德特·范德瓦尔登的Moderne AlgebraⓉ（《现代代数》）第一卷。1934年2月，她开始在普林斯顿高等爱德华·斯图迪研究所每周讲课。在1934年3月6日给哈塞的一封信中，她写道：-

我已经从表示模、带算子的群开始……；普林斯顿今年冬天将首次接受代数处理，而且是彻底的处理。我的听众主要是研究员，除了亚伯拉罕·阿德里安·艾伯特和哈里·范迪弗，但我开始意识到我必须小心；毕竟，他们本质上习惯于显式计算，而我已经用我的方法赶走了几个人。

埃米·诺特于1934年夏天返回德国。在那里她最后一次见到了她的兄弟弗里茨，并在前往哥廷根之前拜访了汉堡的埃米尔·阿廷。1980年，埃米尔·阿廷的妻子回忆了埃米·诺特的访问[35]：-

现在我最清晰地记得的一件事是乘坐汉堡地铁的经历，那是汉堡的地下铁。我们在研究所接上了埃米·诺特，她和埃米尔·阿廷立刻开始谈论数学。当时是理想理论，他们开始谈论Ideal、Führer和Gruppe、Untergruppe，整个车厢的人突然都竖起了耳朵。[每个德语名词都有数学和政治双重含义。]我吓得要死——我想，天哪，接下来会发生什么，有人会逮捕我们。当然，那是1934年，等等。但埃米·诺特完全没注意到，她非常大声、非常兴奋地说话，声音越来越大，一直冒出“Führer”和“Ideal”。她充满活力，总是说得又快又响。

她回到美国，她在布林莫尔学院的访问教授职位又延长了一年。她继续在普林斯顿每周讲课，理查德·布饶尔现在也到了那里。讲课后，她喜欢与赫尔曼·外尔、奥斯瓦尔德·维布伦和理查德·布饶尔谈论数学。

埃米·诺特的去世突然而意外。1935年4月，医生发现她有一个肿瘤。两天后他们做了手术，发现更多肿瘤，他们认为这些是良性的，没有切除。手术似乎很成功，三天内她的病情有所好转。然而，第四天她突然倒下，发起了高烧。当天晚些时候她去世了。

赫尔曼·外尔在他的纪念演讲[69]中说：-

她对代数学的重要性不能完全从她自己的论文中看出，她具有极大的激发能力，她的许多建议只在她学生和合作者的著作中才成形。

在[67]中，巴特尔·伦德特·范德瓦尔登写道：-

埃米·诺特，数、函数和运算之间的关系变得透明，易于推广，并且只有在它们与任何特定对象分离并归结为一般概念关系之后才富有成效。

尽管她一生中因其取得的显著进展而鲜获认可，但在她去世后，人们以多种方式纪念她。月球上的一座环形山以她命名。她家乡的一条街道以她命名，她曾就读的学校现在以埃米·诺特学校命名。各种组织以埃米·诺特的名字设立奖学金和讲座。

Emmy Noether's father, Max Noether, was a distinguished mathematician and a professor at Erlangen but he came from a family of wholesale hardware dealers. Her mother was Ida Amalia Kaufmann (1852-1915), from a wealthy Cologne family. Both Emmy's parents were of Jewish origin and the reader may be surprised at this since Noether is not a Jewish name. We should explain, therefore, how this came about and, at the same time, give some information on Emmy Noether's ancestors. Max Noether's paternal grandfather was Elias Samuel, the founder of a business in Bruchsal. Elias had nine children, one being a son Hertz Samuel. In 1809 the State of Baden made the Tolerance Edict which required Jews to adopt Germanic names. Elias Samuel chose the surname Nöther, becoming Elias Nöther, but also changed the given names of his children, giving Hertz the name Hermann. When he was eighteen years old, Hermann Nöther left his home town of Bruchsal and studied theology at the University of Mannheim. Then in 1837, together with his brother Joseph, he set up a wholesale business in iron hardware. Hermann Nöther and his wife Amalia had five children, the third of which was Max. The two children older than Max were Sarah (born 6 November 1839) and Emil. It is worth noting at this point that the Nöther iron-wholesaling business remained a family firm for exactly one hundred years, until the Nazis removed Jewish families from their own businesses in 1937. One other comment is necessary at this point. Although the family name was chosen to be Nöther by Max's grandfather, Max and his family always used the form Noether (except on Max's wedding certificate where the form Nöther appears).

Emmy was the eldest of her parents' four children, the three younger children being boys. Alfred Noether (1883-1918) studied chemistry and was awarded a doctorate from Erlangen in 1909. However, his career was short since he died nine years later. Fritz Noether (1884-1941) became an applied mathematician. However, as a Jew he was unable to work and left Germany in 1937. He was appointed as a professor at the University of Tomsk in the Soviet Union but accused of anti-Soviet acts he was sentenced to death and shot. He was found not guilty by the Supreme Court of the Soviet Union in 1988. Gustav Robert Noether (1889-1928) had bad health all his life. He was mentally handicapped, spent most of his life in an institution and died young. The first school that Emmy attended was on Fahrstrasse. Auguste Dick writes [5]:-

Emmy did not appear exceptional as a child. Playing among her peers in the schoolyard on Fahrstrasse she probably was not especially noticeable - a near-sighted, plain-looking little girl, though not without charm. Her teachers and classmates knew Emmy as a clever, friendly, and likeable child. She had a slight lisp and was one of the few who attended classes in the Jewish religion.

After elementary school, Emmy Noether attended the Städtische Höhere Töchter Schule on Friedrichstrasse in Erlangen from 1889 until 1897. She had been born in the family home at Hauptstrasse 23 and lived there until, in the middle of her time at high school, in 1892, the family moved to a larger apartment at Nürnberger Strasse 32. At the high school she studied German, English, French, arithmetic and was given piano lessons. She loved dancing and looked forward to parties with children of her father's university colleagues. At this stage her aim was to become a language teacher and after further study of English and French she took the examinations of the State of Bavaria and, in 1900, became a certificated teacher of English and French in Bavarian girls schools. She was awarded the grade of "very good" in the examinations, the weakest part being her classroom teaching.

However Noether never became a language teacher. Instead she decided to take the difficult route for a woman of that time and study mathematics at university. Women were allowed to study at German universities unofficially and each professor had to give permission for his course. Noether obtained permission to sit in on courses at the University of Erlangen during 1900 to 1902. She was one of only two female students sitting in on courses at Erlangen and, in addition to mathematics courses, she continued her interest in languages being taught by the professor of Roman Studies and by an historian. At the same time she was preparing to take the examinations which allowed a student to enter any university. Having taken and passed this matriculation examination in Nürnberg on 14 July 1903, she went to the University of Göttingen. During 1903-04 she attended lectures by Karl Schwarzschild, Otto Blumenthal, David Hilbert, Felix Klein and Hermann Minkowski. Again she was not allowed to be a properly matriculated student but was only allowed to sit in on lectures. After one semester at Göttingen she returned to Erlangen.

At this point the rules were changed and women students were allowed to matriculate on an equal basis to the men. On 24 October 1904 Noether matriculated at Erlangen where she now studied only mathematics. In 1907 she was granted a doctorate after working under Paul Gordan. The oral examination took place on Friday 13 December and she was awarded the degree 'summa cum laude'. Hilbert's basis theorem of 1888 had given an existence result for finiteness of invariants in $n$ variables. Gordan, however, took a constructive approach and looked at constructive methods to arrive at the same results. Noether's doctoral thesis followed this constructive approach of Gordan and listed systems of 331 covariant forms. Colin McLarty writes that [39]:-

... her dissertation of 1908 with Gordan pursued a huge calculation that had stumped Gordan forty years before and which Noether could not complete either. So far as I know no one has ever completed it or even checked it as far as she went. It was old-fashioned at the time, a witness to the pleasant isolation of Erlangen, and made no use of Gordan's own work building on Hilbert's ideas.

Having completed her doctorate the normal progression to an academic post would have been the habilitation. However this route was not open to women so Noether remained at Erlangen, helping her father who, particularly because of his own disabilities, was grateful for his daughter's help. Noether also worked on her own research, in particular she was influenced by Ernst Fischer who had succeeded Gordan to the chair of mathematics when he retired in 1911. Noether wrote about Fischer's influence:-

Above all I am indebted to Mr E Fischer from whom I received the decisive impulse to study abstract algebra from an arithmetical viewpoint, and this remained the governing idea for all my later work.

Fischer's influence took Noether towards Hilbert's abstract approach to the subject and away from the constructive approach of Gordan. Now this was very important to her development as a mathematician for Gordan, despite his remarkable achievements, had his limitations. Noether's father, Max Noether, said of Gordan (see [3]):-

Gordan was never able to do justice to the development of fundamental concepts; even in his lectures he completely avoided all basic definitions of a conceptual nature, even that of the limit.

Noether's reputation grew quickly as her publications appeared. In 1908 she was elected to the Circolo Matematico di Palermo, then in 1909 she was invited to become a member of the Deutsche Mathematiker-Vereinigung and in the same year she was invited to address the annual meeting of the Society in Salzburg. She gave the lecture Zur Invariantentheorie der Formen von n Variabeln Ⓣ. In 1913 she lectured in Vienna, again to a meeting of the Deutsche Mathematiker-Vereinigung. Her lecture on this occasion was Über rationale Funktionenkörper Ⓣ. While in Vienna she visited Franz Mertens and discussed mathematics with him. One of Merten's grandsons remembered Noether's visit (see [5]):-

... although a woman, [she] seemed to me like a Catholic chaplain from a rural parish - dressed in a black, almost ankle-length and rather nondescript, coat, a man's hat on her short hair ... and with a shoulder bag carried crosswise like those of the railway conductors of the imperial period, she was rather an odd figure.

During these years in Erlangen she advised two doctoral students who were both officially supervised by her father. These were Hans Falckenberg (doctorate 1911) and Fritz Seidelmann (doctorate 1916).

For information on these and Noether's other Ph.D. students see THIS LINK.

In 1915 Hilbert and Klein invited Noether to return to Göttingen. The reason for this was that Hilbert was working on physics, in particular on ideas on the theory of relativity close to those of Albert Einstein. He decided that he needed the help of an expert on invariant theory and, after discussions with Klein, they issued the invitation. Van der Waerden writes [68]:-

She came and at once solved two important problems. First: How can one obtain all differential covariants of any vector or tensor field in a Riemannian space? ... The second problem Emmy investigated was a problem from special relativity. She proved: To every infinitesimal transformation of the Lorentz group there corresponds a Conservation Theorem.

This result in theoretical physics is sometimes referred to as Noether's Theorem, and proves a relationship between symmetries in physics and conservation principles. This basic result in the theory of relativity was praised by Einstein in a letter to Hilbert when he referred to Noether's penetrating mathematical thinking. Of course, she arrived in Göttingen during World War I. This was a time of extreme difficulty and she lived in poverty during these years and politically she became a radical socialist. However, they were extraordinarily rich years for her mathematically. Hermann Weyl, in [69] writes about Noether's political views:-

During the wild times after the Revolution of 1918, she did not keep aloof from the political excitement, she sided more or less with the Social Democrats; without being actually in party life she participated intensely in the discussion of the political and social problems of the day. ... In later years Emmy Noether took no part in matters political. She always remained, however, a convinced pacifist, a stand which she held very important and serious.

Hilbert and Klein persuaded her to remain at Göttingen while they fought a battle to have her officially on the Faculty. In a long battle with the university authorities to allow Noether to obtain her habilitation there were many setbacks and it was not until 1919 that permission was granted and she was given the position of Privatdozent. During this time Hilbert had allowed Noether to lecture by advertising her courses under his own name. For example a course given in the winter semester of 1916-17 appears in the catalogue as:-

Mathematical Physics Seminar: Professor Hilbert, with the assistance of Dr E Noether, Mondays from 4-6, no tuition.

At Göttingen, after 1919, Noether moved away from invariant theory to work on ideal theory, producing an abstract theory which helped develop ring theory into a major mathematical topic. Idealtheorie in Ringbereichen Ⓣ (1921) was of fundamental importance in the development of modern algebra. In this paper she gave the decomposition of ideals into intersections of primary ideals in any commutative ring with ascending chain condition. Emanuel Lasker (who became the world chess champion) had already proved this result for a polynomial ring over a field. Noether published Abstrakter Aufbau der Idealtheorie in algebraischen Zahlkörpern Ⓣ in 1924. In this paper she gave five conditions on a ring which allowed her to deduce that in such commutative rings every ideal is the unique product of prime ideals.

In the same year of 1924 B L van der Waerden came to Göttingen and spent a year studying with Noether. After returning to Amsterdam van der Waerden wrote his book Moderne Algebra Ⓣ in two volumes. The major part of the second volume consists of Noether's work. From 1927 onwards Noether collaborated with Helmut Hasse and Richard Brauer in work on non-commutative algebras. They wrote a beautiful paper joint paper Beweis eines Hauptsatzes in der Theorie der Algebren Ⓣ which was published in 1932. In addition to teaching and research, Noether helped edit Mathematische Annalen. Much of her work appears in papers written by colleagues and students, rather than under her own name.

Further recognition of her outstanding mathematical contributions came with invitations to address the International Congress of Mathematicians at Bologna in September 1928 and again at Zürich in September 1932. Her address to the 1932 Congress was entitled Hyperkomplexe Systeme in ihren Beziehungen zur kommutativen Algebra und zur Zahlentheorie Ⓣ. In 1932 she also received, jointly with Emil Artin, the Alfred Ackermann-Teubner Memorial Prize for the Advancement of Mathematical Knowledge. In April 1933 her mathematical achievements counted for nothing when the Nazis caused her dismissal from the University of Göttingen because she was Jewish. She received no pension or any other form of compensation but, nevertheless, she considered herself more fortunate than others. She wrote to Helmut Hasse on 10 May 1933 (see for example [5]):-

Many thanks for your dear compassionate letter! I must say, though, that this thing is much less terrible for me than it is for many others. At least I have a small inheritance (I was never entitled to a pension anyway) which allows me to sit back for a while and see.

Weyl spoke about Noether's reaction to the dire events that were taking place around her in the address he gave at her funeral:-

You did not believe in evil, indeed it never occurred to you that it could play a role in the affairs of man. This was never brought home to me more clearly than in the last summer we spent together in Göttingen, the stormy summer of 1933. In the midst of the terrible struggle, destruction and upheaval that was going on around us in all factions, in a sea of hate and violence, of fear and desperation and dejection - you went your own way, pondering the challenges of mathematics with the same industriousness as before. When you were not allowed to use the institute's lecture halls you gathered your students in your own home. Even those in their brown shirts were welcome; never for a second did you doubt their integrity. Without regard for your own fate, openhearted and without fear, always conciliatory, you went your own way. Many of us believed that an enmity had been unleashed in which there could be no pardon; but you remained untouched by it all.

For a version of Weyl's speech see THIS LINK.

She accepted a one-year visiting professorship at Bryn Mawr College in the USA and in October 1933 sailed to the United States on the ship Bremen to take up the appointment. She had hoped to delay accepting the invitation since she would have liked to have gone to Oxford in England but it soon became clear that she had to leave quickly. At Bryn Mawr she was made very welcome by Anna Johnson Pell Wheeler who was head of mathematics. Noether ran a seminar during the winter semester of 1933-34 for three students and one member of staff. They worked through the first volume of van der Waerden's Moderne Algebra Ⓣ. In February 1934 she began giving weekly lectures at the Institute for Advanced Study, Princeton. In a letter to Hasse, dated 6 March 1934, she wrote:-

I have started with representation modules, groups with operators ...; Princeton will receive its first algebraic treatment this winter, and a thorough one at that. My audience consists mostly of research fellows, besides Albert and Vandiver, but I'm beginning to realise that I must be careful; after all, they are essentially used to explicit computation and I have already driven a few of them away with my approach.

Noether returned to Germany in the summer of 1934. There she saw her brother Fritz for what would be the last time, and visited Artin in Hamburg before going on to Göttingen. In 1980 Artin's wife recalled Noether's visit [35]:-

Now the one thing I remember most vividly is the trip on the Hamburg Untergrund, which is the subway in Hamburg. We picked up Emmy at the Institute, and she and Artin immediately started talking mathematics. At that time it was Idealtheorie, and they started talking about Ideal, Führer, and Gruppe, and Untergruppe, and the whole car suddenly started pricking up their ears. [Each of the German nouns has both mathematical and political meanings.] And I was frightened to death - I thought, my goodness, next thing's going to happen, somebody's going to arrest us. Of course, that was in 1934, and all. But Emmy was completely oblivious, and she talked very loudly and very excitedly, and got louder and louder, and all the time the "Führer" came out, and the "Ideal." She was very full of life, and she constantly talked very fast and very loud.

She returned to the United States where her visiting professorship at Bryn Mawr had been extended for a further year. She continued her weekly lectures at Princeton where Richard Brauer had now arrived. After her lectures she enjoyed talking about mathematics with Weyl, Veblen and Brauer.

Noether's death was sudden and unexpected. In April 1935 doctors discovered that she had a tumour. Two days later they operated, finding further tumours which they believed to be benign and did not remove. The operation seemed a success and for three days her condition improved. However, on the fourth day she suddenly collapsed and developed a very high temperature. She died later that day.

Weyl in his Memorial Address [69] said:-

Her significance for algebra cannot be read entirely from her own papers, she had great stimulating power and many of her suggestions took shape only in the works of her pupils and co-workers.

In [67] van der Waerden writes:-

For Emmy Noether, relationships among numbers, functions, and operations became transparent, amenable to generalisation, and productive only after they have been dissociated from any particular objects and have been reduced to general conceptual relationships.

Although she received little recognition in her lifetime considering the remarkable advances that she made, she has been honoured in many ways following her death. A crater on the moon is named for her. A street in her hometown is named for her and the school she attended is now named the Emmy Noether School. Various organisations name scholarships and lectures after Emmy Noether.