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 ):-
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  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 :-
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 ):-
... 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 :-
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 .
These two topics on which Eisenstein worked were both strongly motivated by Gauss's Disquisitiones arithmeticae Ⓣ and the paper  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  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 :-
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 :-
... developed his own independent analytic theory of elliptic functions, based on the technique of summing certain conditionally convergent series.Kronecker wrote (see for example ):-
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 , 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 :-
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  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 ...
Article by: J J O'Connor and E F Robertson