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Hurwitz studied the genus of the Riemann surface and worked on how class number relations could be derived from modular equations.
Hurwitz was first taught by Schubert and then was an undergraduate at the University of Berlin where he attended classes by Kummer, Weierstrass and Kronecker.
He then went with Klein to Leipzig where his Ph. D. was supervised by Klein. He received the degree in 1880 with a Ph. D. dissertation on modular functions.
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.
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Tell me about Hurwitz's work on set theory
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