In 1885 Fricke took the examinations at Leipzig to qualify him to teach mathematics in German gymnasiums and, in the same year, he submitted his doctoral thesis Über Systeme elliptischer Modulfunktionen von niederer Stufenzahl Ⓣ. He then returned to Braunschweig, the city in which he was brought up, taking up an appointment as a mathematics teacher at a Gymnasium there. Over the next six years, Fricke taught at two different gymnasiums in Braunschweig but was also given leave of absence to act as tutor to the sons of Albrecht, the Prince Regent of the Duchy of Braunschweig. The boys he taught were Friedrich Heinrich Albrecht (born 1874) and Joachim Albrecht (born 1876).
In fact Fricke's appointment as a tutor proved significant for his future career, for it gave him lots of free time to devote to mathematics. He had kept in touch with Klein, who had moved to a chair in Göttingen in 1886, and they began a collaboration writing the treatise Vorlesungen über die Theorie der elliptischen Modulfunctionen Ⓣ. The first volume of this two-volume work appeared in 1890 published by B G Teubner of Leipzig. The second volume was published in 1892. A study of the theory of elliptic modular functions began with Eisenstein's memoir of 1847, and was further developed in the lectures given by Weierstrass on elliptic functions. Klein's school had made major advances in the theory to which Schläfli (1870), Klein himself, von Dyck, Fricke and Hurwitz all contributed. Klein and Fricke set out this theory in their two-volume text which became a classic.
Between the publication of the two volumes, in 1891, Fricke decided to give up his teaching career and seek an academic post. He habilitated at the University of Kiel, becoming a Privatdozent there in 1891. After a year in Kiel, Fricke decided that in order to further his research career he needed to be nearer to Klein so he moved to Göttingen becoming a Privatdozent there in 1892. In 1894 he was appointed to a chair at the Brunswick Polytechnikum in Braunschweig. In fact he succeeded to the chair which had been held for the previous 32 years by Richard Dedekind who had retired on 1 April 1894. Many years later, Fricke, along with Emmy Noether and Øystein Ore, were joint editors of the Collected Works of Richard Dedekind which were published in three volumes in 1930-32. Returning to the year 1894, we note that in that year Fricke married Eleonora Flender, a niece of Felix Klein.
Let us look at the titles of some of Fricke's papers: Arithmetische Entwicklungen zur Theorie der linearen Differentialgleichungen zweiter Ordnung Ⓣ (1891); Über die Moduln der algebraischen Gebilde Ⓣ (1892); Die Discontinuitätsbereiche der Gruppen reeller linearer Substitutionen einer complexen Veränderlichen Ⓣ (1894); Über eine einfache Gruppe von 360 Operationen Ⓣ (1896); Über die Beziehungen zwischen der Zahlentheorie und der Theorie der automorphen Functionen Ⓣ (1897); Über eine einfache Gruppe von 504 Operationen Ⓣ (1898); and Zur Theorie der Poincaréschen Reihen Ⓣ (1900). The authors of  write:-
Fricke was highly respected both as a mathematician and personally, working closely with Klein to develop much of the theory of what are now called Kleinian groups .... He played a leading role in the University administration, being Rektor from 1904-6 and again from 1921-3. His activity extended to state educational affairs where his experience as a school teacher was valued and he held several official posts. These many duties account for the long delay between the appearance of the first and second volumes (in 1897 and 1912 respectively) of the second opus 'Vorlesungenuber die Theorie der automorphen Funktionen' Ⓣ, of which Fricke was really the author, although with much input from Klein. In the final volume, Fricke took the opportunity to use new developments like Cantor's set theory and Brouwer's theory of dimension to solve a number of problems which had been unresolved in the past.Let us look a little more closely at some of the books which Fricke published. Kurzgefasste Vorlesungen über verschiedene Gebiete der höheren Mathematik, mit Berüchsichtigung der Anwendungen Ⓣ was published in 1900 and was reviewed by Henry White  in the following year:-
Two objects are sought in Dr Fricke's timely book: first, to supply a defect in German mathematical literature, a handbook for students who have mastered the elements of analysis and are not yet qualified to read profitably the highly specialized treatises; second, to smooth the way for technical students who discover a taste for the more abstract branches of mathematics. The present volume is confined to analysis and theory of functions, a second is announced as in preparation, to treat of advanced portions of algebra and geometry. ... Most teachers to whom the purpose indicated by Dr Fricke appeals as eminently desirable will find this book full of helpful suggestions, and will moreover find their own interest in the successive topics vigorously stimulated by the occasional reading of a chapter. The second volume will be awaited with not a little of curiosity.In 1915 Fricke published the first volume of Die elliptischen funktionen und ihre anwendungen. Ⓣ He writes in the Introduction:-
That I should adhere in the main to the methods of presentation, the use of the invariant theory, geometric representation, and so forth, which more than thirty years of close scientific companionship with my teacher and friend F Klein have made my own, I may regard as self-evident.L Wayland Dowling reviews this text in :-
The present volume is the first of a series of three which Dr Fricke proposes to write on the elliptic functions and their applications. It appeared in October, 1915, and is devoted to the function theoretic and analytic bases of the theory of elliptic functions. One would naturally expect that a treatise on elliptic functions from the pen of Dr Fricke would follow the lines of thought developed by Klein and his students thirty-odd years ago. Consequently, on turning the pages of the present volume, one is not surprised to be reminded again and again of modes of thought, of formulas, and of geometric diagrams made familiar through the Klein-Fricke Modulfunktionen.The second volume of this work was published in 1922 but the third volume, although planned by Fricke from the beginning of the project, was never published.
Heinrich Weber had published his three-volume Lehrbuch der Algebra Ⓣ in 1895. Weber died in 1913 and, after the last edition of his famous work went out of print, the publisher, F Vieweg & Sohn, invited Fricke to write a treatise on algebra to replace that of Weber. The first volume of Fricke's Lehrbuch der Algebra, subtitled Verfasst mit Benutzung von Heinrich Webers gleichnamigem Buche Ⓣ, was published in 1924. L E Dickson, in the review , writes:-
Professor Fricke is an experienced writer of unusual clearness. In addition to his recent book on elliptic functions, he was joint author with Klein of extended treatises on elliptic modular functions and automorphic functions. These earlier books show that the author has long been familiar with the field covered by the new algebra. The latter will be somewhat simpler to read than Weber's "Algebra".The second volume, subtitled Ausführungen über Gleichungen niederen Grades Ⓣ, was published in 1926 and was also reviewed by L E Dickson :-
Fricke's "Algebra" is a worthy successor to Weber's "Algebra", which it henceforth displaces. The present volume is a very attractive exposition of the modern theory of equations of degrees 5, 6, 7. The approach is quite different from that by Weber and more attractive since extensive use is made of the geometric and function-theoretic methods developed by Klein in his 'Ikosaeder' and books on elliptic modular functions. Fricke's long experience with the latter subject made it easy for him to give a simple authoritative exposition of those portions of it which suffice for the transcendental solutions of equations of low degrees.The third volume subtitled AIgebraische Zahlen was published in 1928.
We mentioned above that Fricke was one of the editor's of Dedekind's Complete Works published in 1930. In fact this was not the first Complete Works he had edited, for he had been one of the editors of Felix Klein's Complete Works published in 1921-23.
The authors of  give the following view of Fricke's contributions:-
Robert Fricke ... who was held in high esteem by his contemporaries, was for the most part considered no more than a faithful disciple of his academic teacher, Felix Klein. Thus Fricke's achievements were not only mostly overshadowed by the reputation of Klein, but large parts of his work were regarded as old-fashioned even by the end of his life.Let us give one example of why Fricke's approach could be considered old-fashioned (see  for further details). In his book Lehrbuch der Algebra Fricke, of course, studies fields. However, even though Heinrich Weber had given an abstract definition of a field as early as 1893, Fricke chose not to do so in his book, looking only at number fields. Emmy Noether pointed out to Fricke that one could define a field in an abstract way as a set with two operations defined on it satisfying certain laws. Fricke said he knew of such a definition but did not consider that it had produced significant changes regarding the relevance attached to the concept of a field. He simply chose to acknowledge Emmy Noether's point by putting a footnote to his definition of a field in the first volume of Lehrbuch der Algebra Ⓣ :-
E Steinitz approached the problem in detail in the years 1908-1910 in his work "Algebraic theory of fields" (1910) and showed that it involves a fruitful extension of the original concept of a field of numbers.Fricke made many important contributions to mathematics in addition to his research papers and books. For many years he was involved with the German Mathematical Society (Deutsche Mathematiker-Vereinigung). For example at the meeting in Cassel in September 1903, chaired by Felix Klein, Fricke presented the paper On new methods and new text-books in England. In this talk he "outlined what Professor Perry is endeavoring to do in England and compared the Perry movement in England with a parallel movement in the German universities. He also exhibited copies of English text-books of elementary mathematics." In 1920 Fricke was President of the Deutsche Mathematiker-Vereinigung.
Fricke continued to hold the chair in Braunschweig until his death.
Article by: J J O'Connor and E F Robertson
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