Christian-Albrechts-Universität zu Kiel


Famous scholars from Kiel:

Ernst Steinitz


Mathematician, author of seminal works on algebra, Professor of Mathematics in Kiel from 1920 to 1928.


I.

There was a chair of mathematics1 at Kiel right from the university's foundation. It was not until the formation of the Mathematical Seminar in 1877 when Leo Pochhammer was appointed professor that a second chair for mathematics was created. Both chairs became free shortly after the end of the First World War when Pochhammer retired in 1919. The incumbent of the first chair, Heinrich Wilhelm Jung, took up an appointment in Halle after a short period of teaching which was interrupted by military service. Issai Schur and Hermann Weyl, as well as others, were suggested for the vacant posts. The second chair was awarded to Otto Toeplitz who had been working as a professor without a permanent chair in the Mathematical Seminar since 1913. Ernst Steinitz was appointed to be Jung's successor also in 1920. On 8th February 1920 David Hilbert had written to Otto Toeplitz: "My first suggestion would be Steinitz, who deserves it and is up and coming. You will get on well with him. From a purely scientific point of view I consider him to be the most successful researcher among the persons mentioned"2. As early as 1909 he had recommended Steinitz for the appointment as a professor without a permanent chair in Würzburg with the words, "Steinitz is no longer a very young experienced lecturer of great versatility and who has worked on numbers theory, set theory, polyhedron geometry and analysis situs; he has recently been on the recommended list almost everywhere but has not been appointed due to adverse circumstances. From a personal point of view he is, without doubt, extremely likeable as well as very modest and agreeable"3. He named Felix Hausdorff, Ludwig Bieberbach and Leon Lichtenstein as suitable candidates for the 2nd and 3rd position in the appointments list in Kiel. Richard Neuendorff, previously a lecturer, received the post of professor without a permanent chair which became vacant on the appointment of Toeplitz. Werner Schmeidler, Helmut Hasse and Robert Schmidt lectured at the Mathematical Seminar in the years that followed.


II.

Ernst Steinitz4 came from a widespread Jewish family in Upper Silesia. He was born in Laurahütte (today part of Siemianowice), a Silesian industrial town about six kilometres north of Katowice on 13th June 1871 and was the first child of Sigismund Steinitz (approx. 1845-1889 who worked in the inland waterway industry) and his wife Auguste, neé Cohn (approx. 1850-1906). He had two brothers, Kurt (1872-1929) and Walter (1882-1963). As well as his extraordinary mathematical ability, the young Ernst soon demonstrated an unusual talent for music. His parents sent him to the Silesian Music Conservatory for 13 years where he studied practical and theoretical aspects of the piano. He wrote several piano sonatas and, when he was 17, a piano trio. After graduating from the Friedrich-Gymnasium in Breslau in 1890 he decided to study mathematics but retained his great passion for music throughout his life. There are reports that he could play Beethoven piano sonatas from memory. While students, Heinrich Heesch (1906-1995), Toeplitz and Steinitz had played music together. There is a story, probably originating from Heesch, that at a symphony concert in Kiel Steinitz spontaneously came to the aid of an indisposed soloist and played the Schumann piano concerto5.

After his studies at the University of Breslau in 1890 he moved to Berlin in 1891 where he attended lectures by Ferdinand Georg Frobenius, Leopold Kronecker and Max Planck, among others, returning to Breslau in1893. He was the first person to receive a research prize for the solution of a problem set by the university there for which he received 200 Reichsmarks and the right to receive a doctorate without payment of any fee. He exercised this right in 1894 after presentation of his inaugural dissertation "Über die Construktion der Configurationen n3" (On the Construction of the n3 Configurations) to the Philosophical Faculty. His supervisor was Jacob Rosanes (1842-1922) who originally came from Brody and under whom Otto Toeplitz had also studied for his doctorate in 1905. After his oral dissertation defence he returned to Berlin where he received his "Habilitation" (qualification to teach at a university) in the field of mathematics at the Technical University Berlin-Charlottenburg in 1897. In this year he also became a member of the German Mathematicians Association which had been founded in 1890. He was a lecturer in mathematics in Berlin from 1898 and, from 1901 also for descriptive geometry. He became acquainted with Issai Schur and Edmund Landau at this time. He also formed a friendship with the mathematician Kurt Hensel, a grandson of Fanny Mendelssohn, in whose house he spent many evenings playing chamber music. During a meeting of the German Mathematicians Association he met Otto Toeplitz who had been born in Breslau in 1881. They became close friends, a friendship which continued during the time they spent together in Kiel. On 10th May 1910 he was appointed as professor at the second chair of mathematics at the Technical University of Breslau which had just been founded and where Constantin Caratheodory and Gerhard Hessenberg already taught. He was also received a "Habilitation" at the University of Breslau and, in 1913, he also became an adjunct professor and taught number theory, algebra, complex analysis and polyhedron theory. In 1918 he was also appointed to an honorary professorship at the University of Breslau. Steinitz married his cousin Martha (1875-1942) in 1911 whom he had taught the piano when he was at school and as a student. Their son Erhard was born in 1912.

On 30th April 1920 he commenced his work as Professor of Mathematics at Kiel University. The family initially lived in the Düsternbrooker Weg and later in the Feldstrasse. During his time in Kiel he developed wide-ranging teaching activities. He lectured not only on algebra and polyhedron theory, the areas of his scientific studies, but also on number theory, complex analysis, analysis situs, geometry, vector analysis and mechanics. He also held joint seminars along with Toeplitz and Hasse.

In a letter to Adolf Kneser dated 28th January 1928 Erich Hecke wrote about Steinitz: "He is certainly a very quiet and modest man, but in my opinion and that of many others, he is a truly profound mathematical thinker who has written extremely significant and seminal works"6.

Ernst Steinitz died of inflammation of a heart valve on 29th September 1928. He was cremated on 3rd October 1928 in Lübeck and the urn buried in Breslau. His widow, who returned to Breslau after his death, took her son to Palestine at the beginning of the Nazi period but could not settle there and returned to Breslau. She suffered the terrible fate of European Jewry as she was deported to Theresienstadt by the Nazis and killed in Treblinka in 1942. Their son remained in Palestine but died in 1948.

Although Steinitz was for many years an active member of the German Mathematicians Society, he was honoured with no obituary in the Society's annual reports. Theodor Kaluza was appointed to his chair in 1929. Toeplitz went to Bonn in 1928 but was removed from his position in 1935 and emigrated to Jerusalem in 1939 where he died in 1940. His successor in Kiel was Abraham Adolf Fraenkel who was appointed to a position in Jerusalem in 1928 where he taught from 1929 to 1931 and then from1 933 to his retirement in 1959. He died in 1965.


III.

Ernst Steinitz left behind a substantial and multi-faceted corpus of work which receives significant attention even today. Alongside his main works on theory of fields (1910) and polyhedron theory (published posthumously in 1934) which we deal with in more detail later, his work also includes contributions to module theory (1899, 1912), abelian groups (1901) and conditionally convergent series (1913, 1914, 1916). He wrote the articles in the "Encyklopädie der mathematischen Wissenschaften" (Encyclopaedia of Mathematical Sciences) on projective geometry (1907) and polyhedron and spatial divisions (1922). A short summary of a lecture "On the theory of abelian groups" which Steinitz gave at the German Mathematical Society's 1900 meeting in Aachen is to be found in the annual proceedings of the German Mathematical Society of 1901. He defined a product of isomorphism classes of finite abelian groups, described the structural constants of the algebra they produced and explored the link with the theory of symmetric functions. This work anticipates the underlying definitions and assertions which were presented much later in the theory of Hall algebras. In his two great works of 1912 on "Rechteckige Systeme und Moduln in algebraischen Zahlkörpern" (Rectangular systems and modules in algebraic number fields) Steinitz developed the theory of elementary divisors for matrices composed of algebraic integers. The structure of the modules which were finitely generated about rings of algebraic integers was fully explained. In three papers in the Crelle Journal, Steinitz analysed the "sum range" of conditionally convergent complex series. He reminded readers of Augustin Louis Cauchy's remark that convergence of certain real series is dependent on the arrangement of their members; he also mentioned the proposition of Peter Lejeune Dirichlet that every absolutely convergent series produces the same sum at every arrangement of the members. Steinitz's starting point was Bernhard Riemann's proposition that a conditionally convergent series of real numbers can be rearranged into a convergent series with a predetermined sum, i.e. that the sum range includes all real numbers.

He then went on to write: "It seems obvious to raise the question of the sum range in the complex area"7, and then provides the answer in the sentence: "The sum range of a conditionally convergent complex series is either a straight line or the whole plain. When extending the analysis to complex numbers with n units which are interpreted in the usual way as points on a n dimensional space, the analogous result is produced: the sum range of a conditionally convergent series is always a linear manifold"8. Steinitz used his analyses of series as the reason for setting out the basis of n dimensional geometry as in the theory of convex quantities. Here he formulated and proved the fundamental exchange set known to every student today in abstract form but which had already been described by Hermann Grassmann in his "Ausdehnungslehre" (Extension Theory) (1844). Landau had drawn Steinitz's attention to the fact that conditionally convergent series had been considered in a paper by Paul Lévy in 1905 but that he was not convinced of the accuracy of the results to be found there. On this matter Steinitz wrote: "This prompted me to study this work once more in the greatest detail. It is written in a very brief, indeed erratic manner and often in a language which is not clear. If the reader is unaware of the solution, he will in places hardly be able to guess what is meant. ... However much we must deplore publications which are in such a deficient form that extensive commentaries are necessary if they are to be understood, we must admit that Mr Lévy has largely proved the proposition on the indicated series with usually complex numbers in the quoted work. The situation with regard to the proposition on generally complex numbers is different, as only results are indicated here. This does not indicate that Mr Lévy has not also found a solution for the general case and it indicates even less that he was not able to find it. But one cannot say that this proof is contained in his paper..."9.

In 1910 Steinitz's comprehensive paper on "Algebraische Theorie der Körper" (Algebraic Theory of Fields) appeared in Volume 137 of the "Journal für die reine und angewandte Mathematik" (Journal of Pure and Applied Mathematics, pages 167-309). In his "Notes historiques" of the "Éléments de Mathématique" Bourbaki wrote about this work: "One can say of this that it is the origin of our modern concept of algebra"10. Reinhold Baer and Helmut Hasse wrote in the foreword to the 1930 reprint of the paper: "Steinitz's work "Algebraische Theorie der Körper" [...] has become the starting point for many far-reaching analyses in the field of algebra and arithmetic. In the classic beauty and perfection of form of its presentation in all its details it is not only a landmark in the development of algebraic knowledge but it also remains today an outstanding, even indispensable introduction for everyone who wishes to devote himself to the field of detailed studies of more modern algebra".11

In Volume 350 (1984) the publishers of "Journal für die reine und angewandte Mathematik" showed a picture of Steinitz with the remark, "75 years ago Steinitz wrote his paper "Algebraische Theorie der Körper" which appeared in Volume 137 of this Journal. The particular significance which Steinitz's work has achieved for the development of modern algebra is based not only on the information it contains on the structural composition of bodies but also because it also primarily contains an advance in methodology: the systematic analysis of abstract algebraic structures based on axiom systems and set theory is to be found for the first time in Steinitz's work."12


IV.

The original problem of algebra is the solution of (algebraic) equations and equation systems. In this process the quantities which define the task and the quantities which are sought are assumed to be (complex) numbers. At the beginning of the 19th century Niels Henrik Abel and Evariste Galois defined those quantities which can be rationally expressed as a function of the given quantities as "basic fields". However, they had no idea of the totality of these quantities. It was particularly the theory of algebraic numbers which produced a decisive change in the second half of the 19th century. Leopold Kronecker and Richard Dedekind considered systems of quantities which permit the normal algebraic operations of addition and multiplication. Kronecker spoke of "areas of rationality". Dedekind introduced the term "bodies". Steinitz's starting point was Heinrich Weber's 1893 paper. In the introduction to his paper he wrote: "In this paper the term "field" is used in the abstract and general manner as in H. Weber's analyses of the general principles of Galois' equation theory, namely as a system of elements with two operations, addition and multiplication which are subject to the associative and cumulative laws, are linked by distributive law and which permit unlimited and unambiguous reversals."3

He goes on: "While Weber's objective is a general consideration of Galois's theory, the concept of fields is the prime focus of interest for us. Gaining an overview of all possible body types and defining their inter-relationships can be considered as the subject of this work."14 In a foot-note to the introduction, Steinitz indicated he was prompted to embark on this work by the theory of the fields of the p-adisc numbers described by Kurt Hensel "which cannot be listed among the function and number bodies in the usual meaning of the word".15

The programme he undertook is systematically conducted in the four chapters into which the work is divided. The algebraic, infinite algebraic and then the transcendental extensions are considered from the principles upwards. Starting from the axioms, the theory is developed with abstract algebra using terms and theorems taken from quantity theory. Many of the terms with which we are familiar, such as "characteristic", "normal", "separable" were introduced and investigated in many different ways. He used these to explain the premises necessary for the validity of the main theorem of Galois' theory. In the third chapter Steinitz provides a proof for the existence and unambiguity of the algebraic shell of a body. In this proof he used Zermelo's well-ordering theorem which was based on what was then the very disputed axiom of choice. On this point he remarked in the introduction, "Many mathematicians still take a negative view of the principle of choice. However, with growing insight that there are issues in mathematics which cannot be resolved without this principle, opposition to this principle is likely to disappear."16


V.

In the winter semesters of 1921/22 and 1923/24 Steinitz lectured on polyhedron theory which had repeatedly absorbed him. There was in his estate an incomplete manuscript (the whereabouts of which is unfortunately unknown) with a comprehensive account of this theory. "There could be no doubt that the work he left behind must be made accessible to the world of mathematics, not only out of respect for the name of STEINITZ but particularly because of its inner quality and singularity"17, wrote Hans Rademacher in his foreword to the book he published in 1934 under the title "Ernst Steinitz: Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie" (Ernst Steinitz: Lectures on the Theory of the Polyhedron along with Elements of Topology) as Volume XLI of the "Grundlehren der mathematischen Wissenschaften" (Basic Theories of Mathematical Sciences). "The manuscript consisted of two thin quarto note-books covered closely with Steinitz's handwriting and of a large number of orderly-arranged individual sheets which were written as he dictated."18 According to Rademacher there were significant gaps in the manuscript and no scheme whatsoever for an overall design for the work. He wrote: "All that there as the basis for the publication was the encyclopaedia article "Polyeder und Raumaufteilung" (Polyhedrons and Spatial Division) which in no way repeated only what was already known; in many places there was Steinitz's own research in outline form but without mentioning the author; there were also Steinitz's hand-written notes to his own Kiel lectures mentioned above. [...] According to the overall layout of the book, convex polyhedrons and their topological types were clearly intended to be the book's subject."19

The work is divided into three large sections. In the first section Steinitz provides a "Historic review of the development of polyhedron theory". He wrote: "There is no uniform definition of the concept of the polyhedron, and it would also not be appropriate to wish to identify one such definition. [...] According to the older explanations, a polyhedron is a body limited by a finite number of even faces; at a later date only these boundaries themselves were often called polyhedrons".20 Steinitz discussed the polyhedron theorems of Euler and Cauchy in great detail, described Legendre's definition of the constant number of a polyhedron and formulated "the general problem of the combinative compilation of the types of complex polyhedrons".

The second chapter is dedicated to the analysis of what are called "polyhedron complexes" and their topological equivalence. In the 3rd chapter "Polyhedrons in the narrower sense" are discussed. Steinitz introduced a combinatory definition of the polyhedron concept. Such polyhedrons are particularly designated as "K polyhedrons". It is easy to understand that every convex polyhedron is a K polyhedron. The converse of this statement is stated to be a "fundamental theorem of the convex type": every K polyhedron can take geometric shape as a convex polyhedron.

Three different proofs for this theorem are provided in the third section of the book. The first proof uses analytical-geometrical methods, whereas the two other proofs are purely geometrical in which express reference is made to Hilbert's "Fundamentals of Geometry" of 1899. Branko Grünbaum wrote in his 1967 monograph "Convex Polytopes", "It is remarkable how relatively unknown an important result may be even if it is the main topic of a monograph published in one of the best-known series"21. The theorem is recognised as one of the main outcomes of polyhedron theory. In the dedication of his book Branko Grünbaum wrote: "In humble homage dedicated to the memory of the outstanding geometer Ernst Steinitz"22.

Ernst Steinitz worked for barely a decade in Kiel. The University has every reason to remember with thanks this man whose influential work places him without doubt among the first rank of mathematicians of the early 20th century. The Mathematics Department of Kiel University has, since May 2007, called its large auditorium the STEINITZ AUDITORIUM.


Prof. Dr. Karsten Johnsen
Mathematisches Seminar




Notes


1      Zur Geschichte der Mathematik in Kiel siehe
  1. Winfried Scharlau (Hrsg.), Mathematische Institute in Deutschland 1800 – 1945, Braunschweig 1990, p. 183 - 188.
  2. Jürgen Schönbeck, Mathematik. In: Geschichte der Christian-Albrechts-Universität Kiel, Band 6, Neumünster 1968, p. 9 – 58.
  3. Wolfgang Lübke, Zur Geschichte des Mathematischen Unterrichts an der Christian-Albrechts-Universität zu Kiel seit 1665. Kiel 1961. Wissenschaftliche Prüfungsarbeit zum Fach Mathematik, UB Kiel Qa 6124.
2 David Hilbert, Brief an Otto Toeplitz vom 8.2.1920. Universitäts- und Landesbibliothek Bonn, Toeplitz B: Dokument 47.
3 David Hilbert, Brief an Wilhelm Wien vom 23.4.1909, Deutsches Museum München, Nachlass Wilhelm Wien, Vorl. Nr. 0076, Alt-Signatur NL 056-007.
4 Zur Biographie von Ernst Steinitz siehe u.a. Hans Röhl, Ernst Steinitz, eine Darstellung seines Mathematischen Werkes. Kiel 1962. Wissenschaftliche Prüfungsarbeit zum Fach Mathematik, UB Kiel Film 159.
5 Hans-Günther Bigalke, Heinrich Heesch. Basel 1988, p. 19.
6 Erich Hecke, Brief an Adolf Kneser vom 28.1.1928, HANS SUB Göttingen, Nachlass Adolf Kneser, Cod. Ms. A. Kneser C 4 : 3.
7 Ernst Steinitz, Bedingt konvergente Reihen und konvexe Systeme. In: Journal für die reine und angewandte Mathematik, Bd. 143 (1913). p. 129.
8 ibid p. 129.
9 ibid p. 130.
10 Nicolas Bourbaki, Elemente der Mathematikgeschichte. Göttingen 1971, S. 103.
11 Ernst Steinitz, Algebraische Theorie der Körper. Neu herausgegeben von Reinhold Baer und Helmut Hasse, Berlin 1930, Foreword.
12 Journal für die reine und angewandte Mathematik Band 350 (1984).
13 Ernst Steinitz, Algebraische Theorie der Körper. In : Journal für die reine und angewandte Mathematik, Band 137 (1910), S. 167.
14 ibid p. 167.
15 ibid p. 167.
16 ibid p. 170.
17 Ernst Steinitz, Vorlesungen über die Theorie der Polyeder. Aus dem Nachlass herausgegeben und ergänzt von Hans Rademacher. Grundlehren der Mathematischen Wissenschaften. Bd. XLI, Berlin 1934. Foreword.
18 ibid foreword.
19 ibid foreword.
20 ibid p. 1.
21 Branko Grünbaum, Convex Polytopes, New York 2003, p. 290.
22 ibid p. V

The author thanks Prof. Dr. Wolfgang Gaschütz for his advice and assistance.

Prof. Dr. Karsten Johnsen
Mathematisches Seminar
Ludewig-Meyn-Str. 4, 24118 Kiel
Telefon:+49 431 880-3664
E-Mail: johnsen@math.uni-kiel.de

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