Abstract
In his biography of Emil Artin, Richard Brauer describes the years from 1931–1941 as a time when “Artin spoke through his students and through the members of his mathematical circle” rather than through written publications. This paper explores these seemingly quiet years when Artin immigrated to America and disseminated ideas about algebraic number theory during this time in his collaboration with George Whaples, a young American mathematician who had just completed his Ph.D. at the University of Wisconsin. The main result of their work is the use of the product formula for valuations to come up with an axiomatic characterization of both algebraic number fields and algebraic function fields with a finite field of constants. These two families of fields are exactly the fields for which class field theory is known to hold. We situate their mathematical work in the broader context of algebraic number theory and their lives within the broader historical context.
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References
Archival sources
Artin File, Notre Dame Archives, University of Notre Dame, Notre Dame, Indiana [=NDA]
Artin File, Indiana University Archives, Indiana University, Boomington, Indiana [=IUA]
Artin, Emil, to Helmut Hasse, 27 November, 1930, Cod. Ms H. Hasse, 1:59, nr. 35, [NSUB]
Artin, Emil, to Helmut Hasse, 24 August, 1931, Cod. Ms H. Hasse, 1:59, nr. 39. [NSUB]
Artin, Emil, to Oswald Veblen, IAS, Princeton, 13 February, 1941, [AC, IAS]
Artin, Emil, to Hermann Weyl, 31 January, 1941, [AC, IAS]
Chevalley, Claude, to Helmut Hasse, 20 June, 1935, Cod. Ms. H. Hasse, 1:19. [NSUB]
Ingraham, Mark H., to Oswald Veblen, 11 February, 1941, [AC, IAS]
Kleene, S. C., to Oswald Veblen, 14 February, 1941, [AC, IAS]
Lefschetz, Solomon, to Father John O’Hara, 12 January, 1937, [NDA]
MacDuffee, C. C., to Oswald Veblen, 13 February, 1941, [AC, IAS]
Matchett, Margaret, On the Zeta Function for Ideles, Thesis (Ph.D) -Indiana University, 1946, [IUA]
Niedersächsische Staats-und Universitätsbibliothek Göttingen, Handschriftenabteilung [=NSUB]
O’Hara, John, to H. B. Wells, 11 June, 1938, [NDA]
Secretary of IAS School of Mathematics to Professor Tracy Thomas, 20 June, 1947, [AC, IAS]
The Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton [=AC, IAS]
Weyl, Hermann, to George Whaples, 15 March, 1941, [AC, IAS]
Weyl, Hermann, to Marshall Stone, 12 March, 1942, [AC, IAS]
Weyl, Hermann, to no name given, 21 May, 1942, [AC, IAS]
Whaples, George, Application to IAS, 10 February 1941, [AC, IAS]
Williams, K. P., to Fernandus Payne, 6 April, 1938, [IUA]
Published sources
Archibald Raymond, C. (ed) (1938) Semicentennial Addresses of the American Mathematical Society 1888–1938. American Mathematical Society, New York
Artin Emil. (1930) Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren. Abhandlungen mathematisches Seminar, Universität Hamburg 8: 292–306
Artin Emil. (1931a) Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkörper. Journal für die reine und angewandte Mathematik 164: 1–11
Artin, Emil. 1931b. Einführung in die Theorie der Gammafunktion. (Hamburger mathematische Einzelschriften 11) Leipzig, Berlin: B. G. Teubner. 35 S.
Artin Emil. (1932a) Über Einheiten relativ galoisscher Zahlkörper. Journal für die reine und angewandte Mathematik 167: 153–156
Artin Emil. (1932b) Über die Bewertungen algebraischer Zahlkörper. Journal für die reine und angewandte Mathematik 167: 157–159
Artin, Emil. 1933. Algebraische Zahlentheorie, Lecture Course Hamburg 1933 (Notes taken by E. A. Eichelbrenner, edited by Peter Ullrich, Mitteilungen der Mathematischen Gesellschaft in Hamburg 21/2 (2002), 159–223).
Artin Emil. (1950) The influence of J. H. M. Wedderburn on the Development of Modern Algebra. Bulletin of the American Mathematical Society 56: 65–72
Artin, Emil. 1951. Algebraic numbers and algebraic functions. Notes by I. Adamson. New York: Institute for Mathematics and Mechanics, New York University.
Artin, Emil. 1959. Theory of algebraic numbers. Notes from Lectures at Göttingen 1956/7. Göttingen: Striker.
Artin, Emil. 1965. Collected papers (Ed. Serge Lang and John Tate). New York: Springer.
Artin Emil. (1967) Algebraic numbers and algebraic functions. Gordan and Breach, New York
Artin, Emil and Tate, John T. 1968. Class field theory. New York: W. A. Benjamin, Inc. Advanced Book Classics. Redwood City, CA: Addison-Wesley Publishing Company, Inc.
Artin Emil, Whaples George. (1943) The theory of simple rings. American Journal of Mathematics 65: 87–107
Artin Emil, Whaples George. (1945) Axiomatic characterization of fields by the product formula for valuations. Bulletin of the American Mathematical Society 51: 469–492
Artin Emil, Whaples George. (1946) A note on axiomatic characterization of fields. Bulletin of the American Mathematical Society 52: 245–247
Ash, Mitchell. 2003. Forced migration and scientific change: Steps towards a new overview. In Intellectual migration and cultural transformation: Refugees from national socialism in the english-speaking world, ed. E. Timms and J. Hughes, 241–263. Vienna: Springer.
Brauer Richard. (1967) Emil Artin. Bulletin of the American Mathematical Society 73: 27–43
Brauer Richard, Helmut Hasse, Emmy Noether. (1932) Beweis eines Hauptsatzes in der Theorie der Algebren. Journal für die reine und angewandte Mathematik 167: 399–404
Chevalley Claude. (1930) Sur la théorie des restes normiques. Comptes Rendus de l’Académie des Sciences Paris 191: 426–428
Chevalley Claude. (1931) Relation entre le nombre des classes d’un sous-corps et celui d’un sur-corps. Comptes Rendus de l’Académie des Sciences Paris 192: 257–258
Chevalley Claude. (1932) Sur la structure de la théorie du corps de classes. Comptes Rendus de l’Académie des Sciences Paris 194: 766–769
Chevalley Claude. (1933) Sur la théorie du corps de classes dans les corps finis et le corps locaux. Journal of the Faculty of Science, University of Tokyo 2: 365–476
Chevalley Claude. (1936) Généralisation de la théorie du corps de classes pour les extensions infinies. Journal de Mathématiques Pures et Appliquées. 15: 359–371
Chevalley Claude. (1940) La théorie du corps de classes. Annals of Mathematics 41: 394–418
Chevalley Claude. (1953) Class field theory. Nagoya University, Nagoya
Chevalley Claude, Jacques Herbrand. (1931) Nouvelle démonstration du théorème d’existence en théorie du corps de classes. Comptes Rendus de l’Académie des Sciences Paris 193: 814–815
Cohn Harvey. (1978) A Classical invitation to algebraic numbers and class fields. With two appendices by Olga Taussky. Springer, New York
Dickson, Leonard. 1983. The collected mathematical papers of Leonard Eugene Dickson (Ed. A. Adrian Albert. 6 vols). New York: Chelsea Publishing Com.
Fenster, Della D. 2007. Artin in America (1937–1958): A time of transition. In: Emil Artin (1898–1962) Beiträge zu Leben, Werk und Persönlichkeit, ed. K. Reich and A. Kreuzer, Algorismus 61, Augsburg: Dr. Erwin Rauner Verlag.
Frei, Günther and Peter Roquette. 2008. Emil Artin und Helmut Hasse: Die Korrespondenz 1923–1934 (Ed. and commented by G. Frei and P. Roquette, with contributions of F. Lemmermeyer). Göttingen: Universitätsverlag.
Fujisaki Genjiro. (1958) On the zeta-function of the simple algebra over the field of rational numbers. Journal of the Faculty of Science, University of Tokyo Section 1A, Mathematics 7: 567–604
Galison Peter. (2008) Ten problems in history and philosophy of science. Isis 99: 111–124
Hasse Helmut. (1923a) Über die Äquivalenz quadratischer Formen im Körper der rationalen Zahlen. Journal für die Reine und Angewandte Mathematik 152: 205–244
Hasse Helmut. (1923b) Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen. Journal für die Reine und Angewandte Mathematik 152: 129–148
Hasse Helmut. (1924a) Äquivalenz quadratischer Formen in einem beliebigen algebraischen Zahlkörper. Journal für die Reine und Angewandte Mathematik 153: 184–191
Hasse Helmut. (1924b) Darstellbarkeit von Zahlen durch quadratische Formen in einem beliebigen algebraischen Zahlkörper. Journal für die Reine und Angewandte Mathematik 153: 113–130
Hasse Helmut. (1924c) Symmetrische Matrizen im Körper der rationalen Zahlen. Journal für die Reine und Angewandte Mathematik 153: 12–43
Hasse Helmut. (1950) Kurt Hensel zum Gedächtnis. Journal für die Reine und Angewandte Mathematik 187: 1–13
Hasse, Helmut. 1975. Mathematische Abhandlungen (Ed. Heinrich Wolfgang Leopoldt and Peter Roquette, 3 vols). Berlin: Walter de Gruyter.
Hecke Erich. (1918) Über eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Erste Mitteilung. Mathematische Zeitschrift 1: 357–376
Hecke Erich. (1920) Über eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen, Zweite Mitteilung. Math. Zeitschrift 4((1920): 11–21
Hensel, Kurt. 1913. Zahlentheorie. Berlin and Leipzig: G. J. Göschen’sche Verlagshandlung.
Hey, Käte. 1927. Analytische Zahlentheorie in Systemen hyperkomplexer Zahlen. Thesis Hamburg 1927.
Iyanaga Shokichi. (2006) Travaux de Claude Chevalley sur la théorie du corps de classes: Introduction. Japanese Journal of Mathematics 1: 25–85
Ostrowski Alexander. (1935) Untersuchungen zur arithmetischen Theorie der Körper (Die Theorie der Teilbarkeit in allgemeinen Körpern). Mathematische Zeitschrift 39: 269–320
Reingold, Nathan. 1985. Refugee mathematicians in the United States of America, 1933–1941: Reception and reaction. Annals of Science 38: 313–338.
Rider, Robin. 1984. Alarm and opportunity: Emigration of mathematicians and physicists to Britain and the United States, 1933–1945. Historical Studies in the Physical Sciences 15: 107–176.
Rota Gian-Carlo, Fabrizio Palombi. (2008) Indiscrete Thoughts. Birckhäuser, Boston
Schwermer, Joachim. 2009. Minkowski, Hensel and Hasse—On the beginnings of the local–global principle. In Episodes in the history of modern algebra (1850–1950), ed. Jeremy J. Gray and Karen Hunger Parshall. Rhode Island: American Mathematical Society.
Siegmund-Schultze, Reinhard. 2009. Mathematicians fleeing from Nazi Germany: Individual fates and global impact. Princeton, NJ: Princeton University Press.
Tate, John. 1950. Fourier analysis in number fields and Hecke’s zeta functions, thesis, Princeton 1950, In: Algebraic number theory, ed. J. W. S. Cassels and A. Fröhlich, pp. 305–347. London: Academic Press.
Wedderburn JosephH.M. (1907) On hypercomplex number systems. Proceedings of the London Mathematical Society 6: 77–118
Weil Andre. (1951) Sur la théorie du corps de classes. Journal of the Mathematical Society of Japan 3: 1–35
Weil Andre. (1967a) Review: The collected papers of Emil Artin. Scripta Mathematica 28: 237–238
Weil, Andre. 1967b. Algebraic number theory. Grundlehren Math. Wissenschaften, Bd. 144, Berlin: Springer.
Whaples George. (1942) Non-analytic class field theory and Grunwald’s theorem. Duke Mathematical Journal 9: 455–473
Whaples George. (1965) Review of introduction to quadratic forms by O. T. O’Meara. American Mathematical Monthly 72: 211–212
Zassenhaus Hans. (1964) Emil Artin, his life and his work. Notre Dame Journal of Formal Logic V: 1–9
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Communicated by: Jeremy Gray.
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Dumbaugh, D., Schwermer, J. The collaboration of Emil Artin and George Whaples: Artin’s mathematical circle extends to America. Arch. Hist. Exact Sci. 66, 465–484 (2012). https://doi.org/10.1007/s00407-012-0100-2
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DOI: https://doi.org/10.1007/s00407-012-0100-2