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He then went with Klein to Leipzig where his Ph. D. was supervised by Klein. He received the degree in 1881 with a Ph. D. dissertation on modular functions Grundlagen einer independenten Theorie der elliptischen Modulfunktionen und Theorie der Multiplikatorgleichungen 1. Stufe.
In 1884 Hurwitz accepted an invitation from Lindemann to fill a chair at Königsberg and he was to remain there for 8 years. Here he taught Hilbert and Minkowski, becoming a life long friend of Hilbert.
In 1892 Frobenius left his chair at Eidgenössische Polytechnikum Zürich to return to Berlin and Hurwitz was appointed to the vacant chair at Zurich. Hurwitz remained at Zurich for the rest of his life, unfortunately continually suffering from ill health.
Hurwitz published a paper on a factorisation theory for integer quaternions in 1896 and applied it to the problem of representing an integer as the sum of four squares. A full proof of Hurwitz' ideas appears in a booklet published in the year of his death. This involves studying the ring of integer quaternions in which there are 24 units. He shows that one-sided ideals are principal and introduces prime and primary quaternions.
Hurwitz studied the genus of the Riemann surface. He worked on how to derive class number relations from modular equations. He investigated the automorphic groups of algebraic Riemann surfaces of genus greater than 1, showing that they were finite.
He also studied complex function theory and wrote several papers on Fourier series.
Article by: J J O'Connor and E F Robertson
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List of References (5 books/articles)
| A Quotation
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| A Poster of Adolf Hurwitz
| Mathematicians born in the same country
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| Cross-references to History Topics | The beginnings of set theory |
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JOC/EFR December 1996
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http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Hurwitz.html
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