费迪南·艾森斯坦Gotthold Eisenstein

费迪南·艾森斯坦的父亲是约翰Konstantin Eisenstein，母亲是海伦·波拉克。这个家庭是犹太人，但在他们的第一个孩子艾森斯坦出生之前，他们已从犹太教改宗成为新教徒。他们的家庭并不富裕，因为Johan Eisenstein在普鲁士军队服役八年后，发现很难适应平民生活中的稳定工作。尽管尝试了各种工作，他一生中大部分时间都没有找到成功的职业，尽管在他生命的末期事情确实对他顺利起来。

艾森斯坦一生都饱受健康不佳之苦，但至少他活过了童年，而他的五个兄弟姐妹都没有做到。他们都死于脑膜炎，而艾森斯坦本人也感染了这种疾病，但他幸存下来。这种疾病以及他童年时患的许多其他疾病，肯定对他产生了心理和身体上的影响，他一生都是疑病症患者。他的母亲海伦艾森斯坦在儿子的早期教育中发挥了重要作用。

他写了一部自传，在其中描述了他母亲在他大约两岁时教他字母表的方式，将物体与每个字母联系起来以暗示其形状，比如O像一扇门，K像一把钥匙。他还在这些自传性作品中描述了自己早期的数学天赋（例如见[1]）：-

当我还是个六岁的男孩时，我理解一个数学定理的证明比理解肉必须用刀切而不是用叉子切更容易。

他从小也展现出相当的音乐天赋，一生中弹钢琴并作曲。

他在小学时就有健康问题，但这些可能与他所上的学校有很大关系。当他大约十岁时，他的父母试图通过送他去夏洛滕堡的考尔学院来解决他持续的健康问题，夏洛滕堡是柏林的一个区，直到1920年才并入该市。这所学校采用了几乎军事化的纪律风格和严格的形式化教育方法，这对艾森斯坦的创造性天性毫无帮助。它不仅没有改善他的健康问题，反而产生了相反的效果，除了持续的身体疾病外，他还患上了抑郁症。

1837年，他十四岁时，艾森斯坦进入Friedrich Wilhelm 文理中学（Gymnasium），然后转到柏林的弗里德里希·韦尔德文理中学完成学业。他一进入弗里德里希·威廉文理中学，他的数学才能就得到了老师们的认可，老师们给予了他各种鼓励。然而，他很快就远远超出了学校的数学教学大纲，从十五岁起，他就购买数学书籍自学。他从莱昂哈德·欧拉和约瑟夫·拉格朗日的著作中学习微分和积分。

到他十七岁时，尽管他还在上学，他开始在柏林大学听约翰·彼得·古斯塔夫·勒热纳·狄利克雷和其他数学家的讲座。大约在这个时候，他的父亲在德国找不到满意的工作，去了英国试图寻找更好的生活。艾森斯坦留在柏林上学，越来越专注于数学。他在自传中写到了自己被数学如此吸引的原因：-

除了实际内容之外，数学如此强烈而专一地吸引我的，尤其是处理数学概念时心智过程的特定性质。这种从旧真理推导和发现新真理的方式，定理的非凡清晰性和自明性，思想的巧妙……对我有一种不可抗拒的吸引力。从个别定理开始，我逐渐习惯于更深入地探究它们之间的关系，并将整个理论作为一个单一实体来把握。我就是这样构想出数学美的概念的……

1842年，他买了一本卡尔·弗里德里希·高斯的Disquisitiones arithmeticae Ⓣ（《算术研究》）的法文译本，并且像约翰·彼得·古斯塔夫·勒热纳·狄利克雷一样，他迷上了他在那里读到的数论。1842年夏天，在参加学校毕业考试之前，他和母亲一起前往英国，与正在那里寻求更好生活的父亲会合。在[12]中，Warnecke认为，在这次英国之行期间，艾森斯坦熟悉了应用技术和科学，这激发了他对数学的普遍兴趣，尤其促成了他成为数学家的愿望。

这家人尝试在威尔士和爱尔兰待了一段时间，但艾森斯坦的父亲找不到合适的工作来给他带来满足和经济保障。随着他们从一个地方搬到另一个地方，艾森斯坦阅读Disquisitiones arithmeticaeⓉ（算术研究），并在可能的时候弹钢琴。1843年在爱尔兰期间，艾森斯坦在都柏林遇到了威廉·哈密顿，这是一个他非常愿意定居的城市，而威廉·哈密顿给了他一份自己写的论文，内容是关于尼尔斯·阿贝尔关于不可能求解五次方程的工作。这进一步激励了艾森斯坦开始数学研究。

1843年6月，艾森斯坦随母亲返回德国，母亲此时与父亲分居。艾森斯坦申请参加学校毕业考试，并获准在8月/9月进行。他以数学老师[1]的高度评价毕业：——

他的数学知识远远超出中学课程的范围。他的天赋和热情使人期待他有朝一日将对科学的发展和扩展作出重要贡献。

他的老师Schellbach是对的，而且没过多久他的期望就实现了。艾森斯坦于1843年秋入读柏林大学，并于1844年1月将威廉·哈密顿的论文提交给柏林科学院。与此同时，他向柏林科学院提交了自己关于二元三次型的论文。

此时他正在研究各种主题，包括二次型和三次型、三次剩余的互反定理、素数的二次分拆以及互反律。奥古斯都·利奥波德·克雷勒被任命为艾森斯坦论文的审稿人，凭借他识别年轻数学人才的惯常直觉，奥古斯都·利奥波德·克雷勒立即意识到这里有一个潜在的天才。奥古斯都·利奥波德·克雷勒与Alexander von Humboldt通信，后者也立即注意到了这位极具天赋的年轻人。艾森斯坦于1844年3月会见了von Humboldt。

艾森斯坦的经济状况很差，von Humboldt特意从国王、普鲁士政府和柏林科学院那里为他争取资助。这些资助多少有些勉强，总是短期，来得晚，而且相当不慷慨。如果不是von Humboldt的个人慷慨，艾森斯坦的处境会比实际上更艰难。但艾森斯坦是一个敏感的人，他不乐意接受这些资助，尤其是当他觉得官方资助是勉强给出的时候。当局本应对他们的钱所带来的回报感到满意，因为艾森斯坦在1844年发表了23篇论文和两个问题在Crelle's Journal上。

1844年6月，艾森斯坦前往哥廷根两周，拜访卡尔·弗里德里希·高斯。卡尔·弗里德里希·高斯以极难被折服而闻名，但艾森斯坦在拜访前已将部分论文寄给卡尔·弗里德里希·高斯，卡尔·弗里德里希·高斯对此赞誉有加。此时艾森斯坦正研究多个课题，包括二次型与三次型以及三次剩余互反律。这次拜访极为成功，艾森斯坦在哥廷根结交了一位朋友，即莫里茨·亚伯拉罕·斯特恩。尽管哥廷根在大学第一年便赢得了即时的国际声誉，他却感到抑郁，而这种抑郁在他短暂的一生中只会愈发严重。

恩斯特·爱德华·库默尔安排布雷斯劳大学于1845年2月授予艾森斯坦荣誉博士学位。卡尔·古斯塔夫·雅各布·雅可比也参与了这一荣誉的安排，但艾森斯坦和卡尔·古斯塔夫·雅各布·雅可比并不总是关系融洽，他们的关系非常起伏不定。从1846年到1847年，艾森斯坦研究椭圆函数，在这两年中的第一年，他卷入了与卡尔·古斯塔夫·雅各布·雅可比的优先权之争。他写信给斯特恩解释情况（例如见[1]）：——

……整个麻烦在于，当我得知[卡尔·古斯塔夫·雅各布·雅可比]关于分圆的工作时，我没有立即公开承认他是创始者，而我在卡尔·弗里德里希·高斯的情况下却经常这样做。我在此事上没有这样做，仅仅是我天真无邪的过错。

1847年，艾森斯坦在柏林大学获得教授资格论文（Habilitation），并开始授课。同年，波恩哈德·黎曼听了他关于椭圆函数的讲座，我们在下文评论了当时波恩哈德·黎曼与艾森斯坦之间可能的互动。

到1848年，德意志邦联的状况已很糟糕。失业与歉收引发了不满与骚乱。1848年2月巴黎起义推翻路易-菲利普的消息导致许多邦国爆发革命，柏林也发生了战斗。共和主义与社会主义情绪意味着君主制陷入困境。艾森斯坦参加了一些支持民主的集会，但未扮演任何积极的政治角色。然而，1848年3月19日，在柏林巷战中，有人从艾森斯坦所在的一所房子（尽管并非他自己的房子）向国王的军队开枪，他因此被捕。他在次日获释，但他所受的严厉对待使他本已虚弱的健康急剧恶化。

这次逮捕还有另一个不良副作用，因为它使那些资助他的人相信他同情共和主义，尽管冯·洪堡继续大力支持他，他获得资金却变得困难得多。关于他在此期间所写的数学著作，安德烈·韦伊在[3]中写道：-

任何艾森斯坦的读者都必定意识到，在他短暂的数学生涯中，他一直感到时间紧迫。……他的论文虽然构思出色，但必定是断断续续写成的，细节只在必要时才加以展开；有时一个论述被中断，直到后来才重新拾起。偶尔，奥古斯都·利奥波德·克雷勒让他在整篇论文完成之前就将一部分寄给出版社。人们常常想起埃瓦里斯特·伽罗瓦那句悲剧性的名言“我没有时间”。

尽管有健康问题，艾森斯坦仍接连发表了一篇又一篇关于素数二次分拆和互反律的论文。他获得了许多荣誉，例如，卡尔·弗里德里希·高斯提名艾森斯坦参选哥廷根科学院，他于1851年当选。1852年初，应约翰·彼得·古斯塔夫·勒热纳·狄利克雷的请求，艾森斯坦当选为柏林科学院。

艾森斯坦因肺结核去世，年仅29岁。他的伟大支持者亚历山大·冯·洪堡当时已83岁，在墓地跟随艾森斯坦的灵柩。他曾成功获得资金，让艾森斯坦前往西西里休养以恢复健康，但为时已晚。

艾森斯坦对数学有三个主要贡献领域，我们已在上文提及。他研究型论，旨在将卡尔·弗里德里希·高斯在Disquisitiones arithmeticaeⓉ（算术研究）中关于二次型理论所得的结果加以推广。他考察高次互反律，旨在推广卡尔·弗里德里希·高斯关于二次互反律的结果，这些结果同样载于Disquisitiones arithmeticae。在这一课题的工作中，艾森斯坦使用了恩斯特·爱德华·库默尔的理想论。恩斯特·爱德华·库默尔与艾森斯坦两人的工作，以及两人在1850年发表的高次互反律工作中存在的竞争，在[7]中有所讨论。

艾森斯坦所研究的这两个主题都受到卡尔·弗里德里希·高斯的Disquisitiones arithmeticaeⓉ（《算术研究》）的强烈启发，而论文[13]讨论了艾森斯坦从学生时代起拥有的这部著作的抄本，该抄本现藏于吉森的数学图书馆。在论文[13]中，安德烈·韦伊考察了艾森斯坦在书中所作的批注，并推测波恩哈德·黎曼在与艾森斯坦的交谈中获得了某些想法，这些想法导致了他关于ζ函数的著名论文。

艾森斯坦做出重大贡献的第三个主题是椭圆函数理论。安德烈·韦伊在[3]中写道：-

艾森斯坦在奠定了椭圆函数理论的基础之后，能够实现他关于这座建筑本身的许多设计，并指出他希望如何将其完成。

尽管这一主题被尼尔斯·阿贝尔和卡尔·古斯塔夫·雅各布·雅可比大大推进，艾森斯坦在1847年关于该主题的论文[10]：-

……发展了他自己独立的椭圆函数分析理论，基于对某些条件收敛级数求和的技巧。

利奥波德·克罗内克写道（例如见[3]）：-

本质上新的观点……特别是关于theta函数的变换理论……由艾森斯坦在基础但鲜被引用的《Beiträge zur Theorie der elliptischen Funktionen》Ⓣ（椭圆函数理论贡献）中引入，该文发表于1847年的奥古斯都·利奥波德·克雷勒杂志，基于完全原创的思想……

事实上，这本书[3]——其第一版于1976年问世，是1974年在普林斯顿高等爱德华·斯图迪研究所讲授课程的结果——正是致力于这一方法。利奥波德·克罗内克采纳了这些主题[3]：-

艾森斯坦的主要主题，经过适当调整，适合大量有趣的变体；……利奥波德·克罗内克的许多最佳工作就由这样的变体组成……

安德烈·韦伊的这本书表明，艾森斯坦的方法对当今正在发展的数学具有重大意义，这是对一位150年前逝世的天才的崇高致敬。一种方法的力量往往通过它为更简单、已被充分理解的情形所带来的洞见而得到体现，安德烈·韦伊确实很好地说明了这一点：-

艾森斯坦表明，他构造椭圆函数的方法完美地适用于更简单的三角函数情形。此外，这一情形不仅为他的理论提供了富有启发性的导引，还为一系列最初由莱昂哈德·欧拉讨论过的结果提供了最简单的证明……

最后，我们引用[10]中关于艾森斯坦的工作在当今的相关性的同一主题：-

从今天的视角回顾，艾森斯坦的数学在我们看来比以往任何时候都更加与时俱进。令我们惊叹的与其说是定理的收获，或成熟理论的创立，不如说是看待事物的方式……

Gotthold Eisenstein's father was Johan Konstantin Eisenstein and his mother was Helene Pollack. The family was Jewish but before Gotthold, who was their first child, was born they had converted from Judaism to become Protestants. Their family were not well off, for Johan Eisenstein, after serving in the Prussian army for eight years, found it hard to adjust to a steady job in civilian life. Despite trying a variety of jobs he did not find a successful occupation for most of his life, although towards the end of his life things did go right for him.

Eisenstein suffered all his life from bad health but at least he survived childhood which none of his five brothers and sisters succeeded in doing. All of them died of meningitis, and Gotthold himself also contracted the disease but he survived it. This disease and the many others which he suffered from as a child certainly had a psychological as well as a physical effect on him and he was a hypochondriac all his life. His mother, Helene Eisenstein, had a major role in her son's early education.

He wrote an autobiography and in it he describes the way that his mother taught him the alphabet when he was about two years old, associating objects with each letter to suggest their shape, like a door for O and a key for K. He also describes his early talent for mathematics in these autobiographical writings (see for example [1]):-

As a boy of six I could understand the proof of a mathematical theorem more readily than that meat had to be cut with one's knife, not one's fork.

He also showed a considerable talent for music from a young age and he played the piano and composed music throughout his life.

While he was at elementary school he had health problems but these may have had a lot to do with the schools which he attended. When he was about ten years old his parents tried to find a solution to his continual health problems by sending him to Cauer Academy in Charlottenburg, a district of Berlin which was not incorporated into the city until 1920. This school adopted an almost military style of discipline and a strict formal approach to education which did nothing for Eisenstein's creative nature. Rather than improve his health problem, it had the opposite effect and in addition to continuing physical illnesses he suffered from depression.

In 1837, when he was fourteen years old, Eisenstein entered the Friedrich Wilhelm Gymnasium then moved to the Friedrich Werder Gymnasium in Berlin to complete his schooling. His mathematical talents were recognised by his teachers as soon as he entered the Friedrich Wilhelm Gymnasium and his teachers gave him every encouragement. However, he soon went well beyond the school syllabus in mathematics and from the age of fifteen he was buying mathematics books to study on his own. He began by learning the differential and integral calculus from the works of Euler and Lagrange.

By the time he was seventeen, although he was still at school, he began to attend lectures by Dirichlet and other mathematicians at the University of Berlin. It was around this time that his father, having failed to find satisfactory employment in Germany, went to England to try to find a better life. Eisenstein remained at school in Berlin becoming more and more devoted to mathematics. He wrote in his autobiography about the reasons that he was so attracted to mathematics:-

What attracted me so strongly and exclusively to mathematics, apart from the actual content, was particularly the specific nature of the mental processes by which mathematical concepts are handled. This way of deducing and discovering new truths from old ones, and the extraordinary clarity and self-evidence of the theorems, the ingeniousness of the ideas ... had an irresistible fascination for me. Beginning from the individual theorems, I grew accustomed to delve more deeply into their relationships and to grasp whole theories as a single entity. That is how I conceived the idea of mathematical beauty ...

In 1842 he bought a French translation of Gauss's Disquisitiones arithmeticae Ⓣ and, like Dirichlet, he became fascinated by the number theory which he read there. In the summer of 1842, before taking his final school examinations, he travelled with his mother to England where they joined his father who was searching for a better life. In [12] Warnecke argues that during this visit to England Eisenstein became familiar with applied technology and science which aroused his interest in mathematics generally and in particular contributed to his desire to become a mathematician.

The family tried spending time in Wales and Ireland but Eisenstein's father could not find the right job to give him satisfaction and financial security. As they moved from place to place Eisenstein read Disquisitiones arithmeticae Ⓣ and played the piano whenever it was possible. While in Ireland in 1843 Eisenstein met Hamilton in Dublin, a city he would have dearly liked to have settled in, and Hamilton gave him a copy of a paper that he had written on Abel's work on the impossibility of solving quintic equations. This further stimulated Eisenstein to begin research in mathematics.

In June 1843 Eisenstein returned to Germany with his mother who separated from his father at this time. Eisenstein applied to take his final school examinations and was allowed to do so in August/September. He graduated with a glowing report from his mathematics teacher [1]:-

His knowledge of mathematics goes far beyond the scope of the secondary school curriculum. His talent and zeal lead one to expect that some day he will make an important contribution to the development and expansion of science.

His teacher, Schellbach, was right and it would not be long before his expectations were fulfilled. Eisenstein enrolled at the University of Berlin in the autumn of 1843 and in January 1844 he delivered Hamilton's paper to the Berlin Academy. At the same time as he submitted to the Berlin Academy his own paper on cubic forms with two variables.

He was working on a variety of topics at this time including quadratic forms and cubic forms, the reciprocity theorem for cubic residues, quadratic partition of prime numbers and reciprocity laws. Crelle was appointed as referee for Eisenstein's paper and, with his usual intuition for spotting young mathematical talent, Crelle immediately realised that here was a potential genius. Crelle communicated with Alexander von Humboldt who also took immediate note of the extraordinarily talented youngster. Eisenstein met von Humboldt in March 1844.

Eisenstein's financial position was poor and von Humboldt went out of his way to obtain grants from the King, the Prussian government, and the Berlin Academy. These were given somewhat grudgingly, always for a short period, arriving late and rather lacking generosity. Had it not been for von Humboldt's personal generosity, Eisenstein would have had a harder time than in fact he had. But Eisenstein was a sensitive person and he was not happy to receive the grants, particularly when he felt that the official ones were given grudgingly. The authorities should certainly have been pleased with the return for their money since Eisenstein published 23 papers and two problems in Crelle's Journal in 1844.

In June 1844 Eisenstein went to Göttingen for two weeks to visit Gauss. Gauss had a reputation for being extremely hard to impress, but Eisenstein had sent some of his papers to Gauss before the visit and Gauss was full of praise. At this time Eisenstein was working on a variety of topics including quadratic and cubic forms and the reciprocity theorem for cubic residues. It was a highly successful visit and Eisenstein made a friend at Göttingen, namely Moritz Stern. Despite the instant international fame that Göttingen achieved while still in his first year at university, he was depressed and this depression would only grow worse through his short life.

Kummer arranged that the University of Breslau award Eisenstein an honorary doctorate in February 1845. Jacobi had also been involved in arranging this honour, but Eisenstein and Jacobi were not always on the best of terms having a very up and down relationship. From 1846 to 1847 Eisenstein worked on elliptic functions and in the first of these years he was involved in a priority dispute with Jacobi. He wrote to Stern explaining the situation (see for example [1]):-

... the whole trouble is that, when I learned of [Jacobi's] work on cyclotomy, I did not immediately and publicly acknowledge him as the originator, while I frequently have done this in the case of Gauss. That I omitted to do so in this instance is merely the fault of my naive innocence.

In 1847 Eisenstein received his habilitation from the University of Berlin and began to lecture. Riemann attended lectures that he gave on elliptic functions in that year and we comment below on possible interaction between Riemann and Eisenstein at this time.

By 1848 conditions were bad in the German Confederation. Unemployment and crop failures had led to discontent and disturbances. The news that Louis-Philippe had been overthrown by an uprising in Paris in February 1848 led to revolutions in many states and there was fighting in Berlin. Republican and socialist feelings meant that the monarchy was in trouble. Eisenstein attended some pro-democracy meetings but did not play any active political role. However, on 19 March 1848, during street fighting in Berlin shots were fired on the King's troops from a house which Eisenstein was in (although it was not his own house) and he was arrested. He was released on the following day but the severe treatment which he had received caused a sharp deterioration in his already delicate health.

The arrest had another bad side effect for it convinced those funding him that he had republican sympathies and it became much harder for him to obtain money although von Humboldt continued to strenuously support him. Writing of his mathematical works written during this period Weil writes in [3]:-

As any reader of Eisenstein must realise, he felt hard pressed for time during the whole of his short mathematical career. ... His papers, although brilliantly conceived, must have been written by fits and starts, with the details worked out only as the occasion arose; sometimes a development is cut short, only to be taken up again at a later stage. Occasionally Crelle let him send part of a paper to the press before the whole was finished. One is frequently reminded of Galois' tragic remark 'Je n'ai pas le temps'.

Despite his health problems Eisenstein published one treatise after another on quadratic partition of prime numbers and reciprocity laws. He was receiving many honours, for example Gauss proposed Eisenstein for election to the Göttingen Academy and he was elected in 1851. Early in 1852, at Dirichlet's request, Eisenstein was elected to the Berlin Academy.

Eisenstein died of pulmonary tuberculosis at the age of 29. His great supporter Alexander von Humboldt, by that time 83 years of age, followed Eisenstein's coffin at the cemetery. He had successfully obtained funds to allow Eisenstein to spend time in Sicily in order to recover his health, but it was too late.

There are three major areas of mathematics to which Eisenstein contributed and we have already mentioned them above. He worked on the theory of forms with the aim of generalising the results obtained by Gauss in Disquisitiones arithmeticae Ⓣ for the theory of quadratic forms. He examined the higher reciprocity laws, with the aim of generalising Gauss's results on quadratic reciprocity, again contained in Disquisitiones arithmeticae. In his work on this topic Eisenstein used Kummer's theory of ideals. The work of both Kummer and Eisenstein, and the rivalry which existed between the two in their work published in 1850 on the higher reciprocity laws, is discussed in [7].

These two topics on which Eisenstein worked were both strongly motivated by Gauss's Disquisitiones arithmeticae Ⓣ and the paper [13] discusses the copy of this work which Eisenstein owned from his days at school which is now in the mathematical library in Giessen. In the paper [13] Weil examines the annotations in the book made by Eisenstein and conjectures that Riemann received ideas in conversations with Eisenstein which led to his famous paper on the zeta function.

The third topic to which Eisenstein made a major contribution was the theory of elliptic functions. Weil writes in [3]:-

Eisenstein, having laid the foundations for a theory of elliptic functions, was able to carry out much of his design for the building itself, and to indicate how he wished it completed.

Although the topic was pushed forward greatly by Abel and Jacobi, Eisenstein's paper on the topic in 1847 [10]:-

... developed his own independent analytic theory of elliptic functions, based on the technique of summing certain conditionally convergent series.

Kronecker wrote (see for example [3]):-

Essentially new points of view ... particularly concerning the transformation theory of theta-functions ...were introduced by Eisenstein in the fundamental but seldom quoted "Beiträge zur Theorie der elliptischen Funktionen" Ⓣ published in Crelle's Journal in 1847, which are based upon entirely original ideas ...

In fact the book [3], the first edition of which appeared in 1976 and was the result of a course given at the Institute for Advanced Study at Princeton in 1974, is devoted to this approach. Kronecker took up these themes [3]:-

Eisenstein's major themes, properly modulated, lend themselves to a large number of interesting variations; ... much of Kronecker's best work consists of such variations ...

This book by Weil shows that Eisenstein's approach is of major importance to the mathematics which is being developed today, a great tribute to a genius who died 150 years ago. Often the power of an approach is illustrated by insight that it adds to simpler well understood cases and indeed this is well illustrated by Weil:-

As Eisenstein shows, his method for constructing elliptic functions applies beautifully to the simpler case of trigonometric functions. Moreover, this case provides not merely an illuminating introduction to his theory, but also the simplest proofs for a series of results, originally discussed by Euler ...

Finally we quote from [10] on the same theme of the relevance of Eisenstein's work today:-

Looking back from today's vantage, Eisenstein's mathematics appear to us more up to date than ever. It is not so much the harvest of theorems, nor the creation of full-fledged theories, but the way of looking at things which amazes us ...