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The Riemann Hypothesis in Characteristic p in Historical Perspective pp 13–37Cite as

The Beginning: Artin’s Thesis

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Part of the book series: Lecture Notes in Mathematics ((HISTORYMS,volume 2222))

Abstract

Emil Artin (1898–1962) was born in Vienna. He was brought up in Reichenberg, a German speaking town in Northern Bohemia belonging to the Austro-Hungarian empire. (The town is now called Liberec, in the Czech Republic). In 1916 he enrolled at the University of Vienna where, among others, he attended a lecture course by Ph. Furtwängler . After one semester of study he was drafted to the army. In January 1919 he entered the University of Leipzig. I have taken this information from Artin’s own hand-written vita that he submitted together with his thesis to the Faculty at Leipzig University. There in June 1921 he obtained his Ph.D. with Herglotz as his thesis advisor.

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Notes

  1. 1.

    The documents for Artin’s Ph.D. are kept in the archive of the University of Leipzig.

  2. 2.

    I am obliged to Frau Dr. Peter from the University of Leipzig for sending me the lecture announcements of the years 1918–1921.

  3. 3.

    I am indebted to Franz Lemmermeyer for pointing out Kühne’s results to me. Kühne (1867–1907) had obtained his Ph.D. in the year 1892 at the University of Berlin. He was a teacher at the technical school in Dortmund. It would be desirable to obtain more biographic information.

  4. 4.

    For the moment, while discussing Artin’s thesis I am denoting the prime ideals of R by gothic letters \(\mathfrak {p}\) since the latin capital P has been used already for prime polynomials in K[x]. Later I will switch again to the notation introduced in Sect. 2.1 where latin letters like P, Q etc. denote primes of F.

  5. 5.

    I have mentioned Kornblum’s paper already, see page 15. The young Kornblum had been a Ph.D. student of Landau in Göttingen. He died in early World War I. The manuscript of his Ph.D. thesis had been almost completed; it was edited and commented by his academic teacher Landau and published 1919 in the newly founded Mathematische Zeitschrift. Artin cites Kornblum, and he points out that his (Artin’s) results on the number of prime polynomials in an arithmetic progression are essentially stronger than Kornblum’s.

  6. 6.

    Constance Reid reports in her book [Rei76] that Herglotz had little contact with his students. If this was the case then it seems that his relation to Artin was an exception.

  7. 7.

    I would like to use this opportunity to point out that it was Blaschke, the first mathematician at the newly founded University of Hamburg, who succeeded to raise this place within a few years to one of the leading mathematical centers in Germany. He did this through a careful Berufungspolitik. The Hamburg Mathematical Seminar in its first decades is a good example that mathematical excellence cannot be created by more money or more positions only, but that the decisive point is to attract excellent people.

  8. 8.

    There is a clash of notation in the mathematical literature, and also in this book. In Analysis the greek letter “π” is used to denote the real number which is half of the circumference of the unit circle. (See formula (3.9) on page 26). In the present context “π” denotes an algebraic number, namely a factor of p in an imaginary quadratic number field. Later when I will discuss Hasse’s theory of complex multiplication in characteristic p the letter “π” will denote the Frobenius operator. (See Sect. 7.4.4). I believe it will always be clear from the context which “π” is meant at the time.

  9. 9.

    The function sin lemn is defined as the inverse function of \(\int _{0}^{x}\frac {dx}{\sqrt {1-x^{4}}}\) , and accordingly cos lemn by means of (3.21).

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  1. Mathematical Institute, Heidelberg University, Heidelberg, Germany

    Peter Roquette

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Roquette, P. (2018). The Beginning: Artin’s Thesis. In: The Riemann Hypothesis in Characteristic p in Historical Perspective. Lecture Notes in Mathematics(), vol 2222. Springer, Cham. https://doi.org/10.1007/978-3-319-99067-5_3

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