After graduating from the Lyceum, Weber entered the University of Heidelberg in 1860 to study mathematics and physics. At this time Heidelberg had some outstanding physicists and mathematicians on the staff who gave courses that Weber attended. These included Gustav Kirchhoff (appointed to Heidelberg in 1854), Robert Bunsen (appointed to Heidelberg in 1852), Hermann von Helmholtz (appointed to Heidelberg in 1858), Otto Hesse (appointed to Heidelberg in 1856), and Moritz Cantor (appointed to Heidelberg in 1853). As was the common practice of German students at this time, Weber spent part of his time studying at a different university. He chose the University of Leipzig in the middle of his studies, spending a semester there in 1862 before returning to Heidelberg to complete his education. He was awarded a doctorate from the University of Heidelberg on 19 February 1863 having been advised by Otto Hesse. He was not required to write a thesis.
In order to become a university teacher, Weber needed to write a further thesis, his habilitation thesis. He went to Königsberg where he studied under Franz Neumann and Friedrich Julius Richelot, who had been a student of Jacobi. Although Jacobi had died over ten years before Weber began his studies at Königsberg, his influence was still strongly felt and it would not be unreasonable to say that Weber, through his teachers at Königsberg, was strongly influenced by Jacobi's style of mathematics. In  Peter Roquette examines why Weber went to Königsberg for his postgraduate study and notes that three of his teachers at Heidelberg, Hesse, Kirchhoff and Helmholtz, came from Königsberg:-
... academic life in Heidelberg around 1863, as far as mathematics and the neighbouring sciences were concerned, was flourishing, and that it was largely dominated by people who came from Königsberg. They knew the stimulating and challenging atmosphere which Königsberg offered to young mathematicians at that time. So it seems natural that young Weber, when he asked his academic teachers where he should go for his postdoctoral studies, was advised to move to Königsberg.There were other students at Königsberg at this time who would become important in the development of mathematics, in particular Albert Wangerin, who studied for his doctorate around the same time as Weber worked for his habilitation, and former students such as Albert Clebsch whose influence was still being felt. In fact Weber mentions the names of nine of his fellow students at Königsberg who went on to high profile careers as scientists.
On 11 August 1866 Weber's habilitation thesis Singuläre Auflösungen partieller Differentialgleichungen erster Ordnung
Together with Richard Dedekind, Weber had been editing Riemann's Collected works and this important book was published in 1876. However, one of the most significant things to happen during his time teaching at Königsberg was in 1880 when two young students, David Hilbert and Hermann Minkowski, enrolled there. At this time Weber was lecturing on number theory, also gave a course on elliptic functions, and ran a seminar on the theory of invariants. All of these courses were attended by the young Hilbert (and probably also by Minkowski). The lecture notes from one of these courses has been preserved, the course on number theory, and Roquette gives an indication of the contents in :-
Weber's course started from the very beginning of number theory, working through congruence calculus, continued fractions, towards quadratic reciprocity. After that, binary quadratic forms are discussed, their classes, their composition, and the analytic class number formulas. Also, the existence of primes in arithmetic progressions, using L-series, is included. In this last part the reader gets the impression that things went quite fast but the proofs are recorded in detail. We conclude that much of the concept of the lecture was based on Gauss's 'Disquisitiones' and Dirichlet's papers. In addition Weber was quite explicit in making use of modern notions, for instance the abstract notion of a group is to be found, as well as what we now call the "main theorem on finite abelian groups", i.e., decomposition of finite abelian groups into cyclic factors (in the context of description of characters).After leaving Königsberg in 1883, Weber was appointed to the Technische Hochschule in Charlottenburg where he spent only one year before he moved again to the Philipps University of Marburg where he was rector of the university during session 1890-91. His next move, in 1892, was to the Georg-August University of Göttingen where he spent three years. His successor at Göttingen was Hilbert. While at Göttingen he published Die allgemeinen Grundlagen der Galois'schen Gleichungstheorie which is the first paper to contain a discussion of an abstract group, either finite or infinite. Weber's final post was to Strasbourg where he was appointed in 1895. It is reasonable to consider why Weber left Göttingen, perhaps the leading mathematical centre in the world at this time, to move to Strasbourg.
Germany had captured Strasbourg after a 50-day siege during the Franco-Prussian war of 1870-71 and annexed the city. After taking control of Alsace-Lorraine, the Germans had reorganised the University of Strasbourg and reopened it as the German Kaiser-Wilhelm University of Strasbourg in 1872. Two mathematics chairs were founded at this time, the first in Mathematics which was filled by Elwin Christoffel and the second in Geometry and Mechanics which was filled by Theodor Reye. Both Christoffel and Reye considered it their patriotic duty to assist in making the University of Strasbourg a German university. Christoffel had been forced to stop teaching in 1892 due to health problems and in 1894 retired leaving the chair vacant. However, Weber did not move to Strasbourg for mathematical reasons, but rather for personal ones. Strasbourg was quite close to Weber's native city of Heidelberg, and one of Weber's daughters, Lina Weber-Holtzmann, had married Heinrich Holtzmann, the professor of theology at Strasbourg, and was living there. Another reason was that his salary was higher in Strasbourg than Göttingen, at least in part because taxes in Alsace were significantly less than those in Prussia. Once he had arrived in Strasbourg he had as colleagues Theodor Reye and the extraordinary professor Adolph Krazer (1858-1926), a student of Friedrich Emil Prym (1841-1919).
Again Weber served as rector of the university in which he was working, taking on this role at Strasbourg during session 1900-01. Note that Weber served as rector three times during his career at three different universities. Weber remained in Strasbourg for the rest of his life and during this time it remained a German city; only at the end of the World War I in 1918 did the city revert to France. Of course, the reader of this biography will have noted the large number of different universities at which Weber worked. In fact he holds the record for the largest number of different university posts by any 19th century German mathematician.
Weber's main work was in algebra, number theory, analysis and applications of analysis to mathematical physics. This seems a contradiction in terms, for we have now almost said that Weber's main work spans the whole spectrum of mathematics. In fact this is not far from the truth for Weber's work was characterised by its breadth across a wide range of topics. To a certain extent this breadth can be attributed to the various influences on Weber from colleagues around him. The applications to mathematical physics certainly grew from working with Franz Neumann in Königsberg. But in Königsberg there was also the Jacobi influence, particularly coming through one of his other teachers Friedrich Richelot, which saw Weber doing important work on algebraic functions.
Perhaps Weber is today remembered for his outstanding text Lehrbuch der Algebra
... the notion of point on an abstract algebraic curve is defined for the first time in history, thus taking a decisive step towards the creation of modern algebraic geometry.The third volume of Lehrbuch der Algebra
... the three men to whom I feel closest in scientific respect.Weber's Lehrbuch der Algebra
A classic from the day of its publication, it is destined to a long and useful career, a monument of honour to its genial author.G A Miller wrote in 1913 :-
Among the advanced text-books on algebra there is probably none which is more favourably known than Weber's "Lehrbuch der Algebra" in three large volumes.For longer extracts from these reviews THIS LINK.
Norbert Schappacher writes 100 years after the book was published:-
[T]his work marks the transition from the late 19th century treatment of algebra to the "modern algebra" whose first full-fledged textbook treatment was going to be van der Waerden's well-known treatise of 1930-31. The third volume would not be called algebra today. It ... contains a classical treatment of elliptic functions, especially their arithmetic theory, along with parts of algebraic number theory and class field theory, as well as a small chapter on differentials of curves in the higher rank case including Riemann-Roch.This third volume of Lehrbuch der Algebra
... he emphasized the decomposition behaviour, as opposed to Hilbert's chief interest in the unramifiedness of the (Hilbert) class field.Among other books by Weber we mention Die Partiellen Differentialgleichungen der Mathematischen Physik
Again extracts from reviews of these books are given at THIS LINK
Bruno Schoeneberg, in , comments on Weber as a teacher:-
Weber was an enthusiastic and inspiring teacher who took great interest in educational questions.Voss writes of Weber's character in :-
On closer contact he opened up without reservation to his friends whom he found through scientific cooperation and common interests.Towards the end of Weber's life a number of tragedies affected him deeply. His wife Emilie died in 1901 and one of his daughters then looked after the household. However, this daughter died in 1909.
Weber received many honours including the publication of a Festschrift compiled by his friends and colleagues to celebrate his 70th birthday on 5 March 1912. He received an honorary doctorate from the University of Kristiana in 1902 at the time of Abel's centenary celebrations. He received an honorary doctorate from the University of Heidelberg in 1912. He was elected to the Göttingen Academy of Sciences (1875), the Berlin Academy of Science (1896), the Munich Academy of Sciences (1903), the Royal Swedish Academy of Sciences, the Uppsala Academy of Sciences, and the Accademia dei Lincei. He was elected president of the German Mathematical Society twice, in 1895 and again in 1904. He was also President of the International Congress of Mathematicians held in Heidelberg in 1904. From 1893 he served as an editor of the Mathematische Annalen.
Article by: J J O'Connor and E F Robertson