In 1831 Kummer was awarded a prize for a mathematical essay he wrote on a topic set by Scherk. In the same year he was awarded his certificate to enable him to teach in schools and, on the strength of his prize winning essay, he was awarded a doctorate. During session 1831-32 Kummer taught a probationary year at the Gymnasium in Sorau where he had been educated. Then he was appointed to a teaching post at the Gymnasium in Liegnitz, now Legnica in Poland.
Kummer held this teaching post in Liegnitz for 10 years. There he taught mathematics and physics and sometimes other topics. Some of his pupils had great ability, and conversely they were extremely fortunate to find a school teacher of Kummer's quality and ability to inspire. His two most famous pupils were Kronecker and Joachimsthal and, under Kummer's guidance, they were undertaking mathematical research while at school.
Kummer himself was undertaking mathematics research while teaching at Liegnitz. He published a paper on hypergeometric series in Crelle's Journal in 1836 and he sent a copy of the paper to Jacobi. This led to Jacobi, and later Dirichlet, corresponding with Kummer on mathematical topics and they soon realised the great potential for the highest level of mathematics that Kummer possessed. In 1839, although still a school teacher, Kummer was elected to the Berlin Academy on Dirichlet's recommendation. Jacobi now realised that he had to find Kummer a university professorship.
In 1840 Kummer married a cousin of Dirichlet's wife. This marriage would only last eight years as his wife sadly died in 1848. However these where years of great achievement for Kummer. In 1842, with strong support from Jacobi and Dirichlet, he was appointed a full professor at the University of Breslau, now Wrocław in Poland. There he quickly established himself as an outstanding university teacher of mathematics and, starting with his move to Breslau, he began to undertake research in number theory. After the death of his wife in 1848, Kummer remarried fairly soon.
During the 22 to 24 February 1848 insurrection in Paris, king Louis-Philippe was overthrown. This resulted in a series of sympathetic revolutions against the governments of the German Confederation. Most of them were tame affairs but in the case of the fighting in Berlin it was bitter and bloody. Nobody could fail to have strong political views at this time and indeed Kummer had strong right wing political views. He supported the constitutional monarchy in the 1848 revolution and was against a republic. When on 13 March 1848 Metternich, a symbol of the establishment, was forced to resign his position in the Austrian Cabinet, the princes quickly made peace with the opposition to prevent republican and socialist experiments like those in France.
In 1855 Dirichlet left Berlin to succeed Gauss at Göttingen. He recommended to Berlin that they offer the vacant chair to Kummer, which indeed they did. Then Kummer played a clever political trick. He wanted Weierstrass as a colleague at Berlin, yet he realised that Weierstrass was a strong candidate for the chair he was leaving vacant in Breslau. Hence he recommended to Breslau that they appoint his former student Joachimsthal. All went according to plan for Kummer and, in 1856, Weierstrass was appointed to Berlin. Kronecker had also been appointed to Berlin in 1855 so Berlin became one of the leading mathematical centres in the world.
In 1861 Kummer and Weierstrass established Germany's first pure mathematics seminar in Berlin. K-R Biermann writes of Kummer's teaching in Berlin in :-
Kummer's Berlin lectures, always carefully prepared, covered analytic geometry, mechanics, the theory of surfaces, and number theory. The clarity and vividness of his presentations brought him great numbers of students - as many as 250 were counted at his lectures. While Weierstrass and Kronecker offered the most recent results of their research in their lectures, Kummer in his restricted himself, after instituting the seminar, to laying firm foundations. In the seminar, on the other hand, he discussed his own research in order to encourage the participants to undertake independent investigations.Explaining Kummer's popularity Biermann writes:-
Kummer's popularity as a professor was based mot only on the clarity of his lectures but on his charm and sense of humour as well. Moreover, he was concerned for the well-being of his students and willingly aided them when material difficulties arose...Surprisingly, given Kummer's outstanding qualities as a teacher of mathematics, he never wrote any textbooks. He did publish mathematical lectures and, of course, many extremely influential mathematical papers.
Having first been elected to the Berlin Academy while still a school teacher, Kummer ended up with high office in the Academy. He was secretary of the Mathematics/ Physics Section of the Academy from 1863 to 1878. He also held high office in the University of Berlin, being Dean in 1857-58 and again in 1865-66. He was rector of the university in 1868-69.
While at Berlin, Kummer supervised a large number of doctoral students including many who went on to hold mathematics chairs at universities, including Bachmann, Cantor, du Bois-Reymond, Gordan, Schönflies and Schwarz. In fact Schwarz became related to Kummer when he married one of his daughters. Kummer also appointed many talented young lecturers including Clebsch, Christoffel and Fuchs. It was Fuchs who succeeded Kummer when he decided to retire in 1883 on the grounds that his memory was failing, although nobody other than Kummer himself ever detected this.
During Kummer's first period of mathematics he worked on function theory. He extended Gauss's work on hypergeometric series, giving developments that are useful in the theory of differential equations. He was the first to compute the monodromy groups of these series.
In 1843 Kummer, realising that attempts to prove Fermat's Last Theorem broke down because the unique factorisation of integers did not extend to other rings of complex numbers, attempted to restore the uniqueness of factorisation by introducing 'ideal' numbers. Not only has his work been most fundamental in work relating to Fermat's Last Theorem, since all later work was based on it for many years, but the concept of an ideal allowed ring theory, and much of abstract algebra, to develop.
The Paris Academy of Sciences awarded Kummer the Grand Prize in 1857 for this work. In fact the prize of 3000 francs was offered for a solution to Fermat's Last Theorem but when no solution was forthcoming, even after extending the date, the Prize was given to Kummer even though he had not submitted an entry for the Prize.
Soon after Kummer was awarded the Grand Prix he was elected to membership of the Paris Academy of Sciences and then, in 1863, he was elected a Fellow of the Royal Society of London. He received numerous other honours in his long career.
Kummer's geometric period was one when he devoted himself to the study of the ray systems that Hamilton had examined, but Kummer treated these problems algebraically. He also discovered the fourth order surface, now named after him, based on the singular surface of the quadratic line complex. The Kummer surface has 16 isolated conical double points and 16 singular tangent planes and was published in 1864.
The three great mathematicians of Berlin, Kummer, Weierstrass and Kronecker were close friends for twenty years as they worked closely and effectively together, However, around 1875 Weierstrass and Kronecker fell out. Kummer continued his friendship with Kronecker but this put a strain on his relation with Weierstrass. Perhaps it is not too surprising that this should happen to these three great mathematical talents, particularly given that Kronecker vigorously attacked personally anyone with whom he had a mathematical difference.
Article by: J J O'Connor and E F Robertson