**Robert Fricke** was a German mathematician who worked in complex analysis, especially on elliptic, modular and automorphic functions.

- Fricke carried out this practice more enthusiastically than most, attending the universities in Göttingen, Zürich, Berlin, and Strassburg between 1880 and 1883.
- Fricke was immediately attracted to Klein's style of mathematics and undertook research for a doctorate under Klein's supervision.
- 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 Ⓣ(On systems of elliptic modular functions of lower-level number).
- 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.
- Fricke's appointment as a tutor proved significant for his future career, for it gave him lots of free time to devote to mathematics.
- 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.
- 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.
- 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.
- Let us look at the titles of some of Fricke's papers: Arithmetische Entwicklungen zur Theorie der linearen Differentialgleichungen zweiter Ordnung Ⓣ(Arithmetic developments in the theory of linear differential equations of second order) (1891); Über die Moduln der algebraischen Gebilde Ⓣ(On the moduli of algebraic structures) (1892); Die Discontinuitätsbereiche der Gruppen reeller linearer Substitutionen einer complexen Veränderlichen Ⓣ(The Discontinuity areas of groups of real linear substitutions of a complex variable) (1894); Über eine einfache Gruppe von 360 Operationen Ⓣ(On a simple group of 360 operations) (1896); Über die Beziehungen zwischen der Zahlentheorie und der Theorie der automorphen Functionen Ⓣ(On the relationship between number theory and the theory of automorphic functions) (1897); Über eine einfache Gruppe von 504 Operationen Ⓣ(On a simple group of 504 operations) (1898); and Zur Theorie der Poincaréschen Reihen Ⓣ(On the theory of Poincare series) (1900).
- 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' Ⓣ(Lectures on the theory of automorphic functions), 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.
- 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.
- In 1915 Fricke published the first volume of Die elliptischen funktionen und ihre anwendungen.
- 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.
- 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 Ⓣ(Textbook of algebra written using Heinrich Weber's book of the same name), was published in 1924.
- 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.
- We mentioned above that Fricke was one of the editor's of Dedekind's Complete Works published in 1930.
- 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.
- 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.
- Fricke made many important contributions to mathematics in addition to his research papers and books.
- 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.
- 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.

Born 24 September 1861, Helmstedt, Germany. Died 18 July 1930, Bad Harzburg, Germany.

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Origin Germany

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive