b. Vienna, 1898; d. Hamburg, Germany, 1962
Number theory; algebra; theory of braids
Born in fin de siècle Vienna to an art dealer father and opera singer mother, Artin was influenced throughout his life by the rich cultural atmosphere of the late Hapsburg Empire. He was, as the algebraist Richard Brauer described him, as much artist as mathematician. After his first semester at the University of Vienna in 1916, Artin was drafted into the Austrian Army, in which he served until the end of World War I. In 1919 he enrolled at the University of Leipzig, and completed his doctorate under the direction of Gustav Herglotz in only two years.
Artin spent the academic year 1921–22 at the mathematically vibrant University of Göttingen, and then moved to the recently opened University of Hamburg. He achieved the rank of full professor in 1926. While at Hamburg, Artin oversaw the work of eleven doctoral students, including Max Zorn and Hans Zassenhaus. Artin’s years at Hamburg were among the most productive of his life.
Artin’s work in CLASS FIELD THEORY [V.28], the subject closest to his heart, led him to a solution of Hilbert’s ninth problem: a proof of the most general law of reciprocity. The aim was to generalize Gauss’s law of quadratic reciprocity and the higher reciprocity laws. Teiji Tagaki’s fundamental results on class field theory had appeared when Artin was a student. Using Takagi’s theory, N. G. Chebotaryov’s 1922 proof of the density theorem (conjectured by FROBENIUS [VI.58] in 1880), and his own theory of L-FUNCTIONS [III.47], Artin established his general law of reciprocity in 1927. Artin’s theorem not only provided the final form of the classical question on reciprocity but it also formed the central result of class field theory. Both Artin’s result and his tools, particularly his L-functions, proved important. Artin posed a conjecture about his L-functions that remains unanswered today. Questions in non-Abelian class field theory also remain open.
In 1926–27, Artin and Otto Schreier developed the theory of formally real fields: fields with the property that –1 cannot be expressed as the sum of two squares (an example being the real numbers). This work formed the basis of Artin’s solution of Hilbert’s seventeenth problem concerning rational functions.
Artin extended Wedderburn’s theory of algebras (“hypercomplex numbers”) to noncommutative rings with chain conditions in 1928. Indeed, the class of such rings called “Artinian rings” is named in his honor.
In 1929 Artin married one of his students, Natalie Jasny. Natalie’s Jewish background and Artin’s personal sense of justice prompted them to leave Germany in 1937. They emigrated to America, where Artin spent a year at Notre Dame University before moving to a permanent position at Indiana University. Artin’s lectures at Notre Dame led to his influential text Galois Theory (1942), which reflected his quest for simplification and his desire to unite different research trends.
At Indiana, Artin began a collaboration with George Whaples of the University of Pennsylvania and introduced the concept of a valuation vector, a notion closely related to the concept of an idèle introduced by Claude Chevalley. This work seemed to revitalize Artin’s mathematical research, and, after something of a hiatus in his written work, he began to publish regularly again.
In 1946, Artin moved to Princeton University. While there, Artin oversaw eighteen of his thirty-one Ph.D. students, including John Tate and Serge Lang. He also returned to his work in the theory of BRAIDS [III.4], a topic that relates questions in topology and group theory. His introduction to the theory of braids that appeared in American Scientist in 1950 reveals Artin’s prowess as a master expositor.
Brauer, R. 1967. Emil Artin. Bulletin of the American Mathematical Society 73:27–43.
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