# Ernst Steinitz

### Quick Info

Born
13 June 1871
Laurahütte, Silesia, Germany (now Siemianowice Śląskie, Poland)
Died
29 September 1928
Kiel, Germany

Summary
Ernst Steinitz worked on the theory of fields.

### Biography

Ernst Steinitz's parents were Sigismund Steinitz (1845-1889) and Auguste Cohn (1850-1906). Sigismund, whose parents were Mosche Laib Steinitz and Philippine Prager, was born in Gleiwitz in Silesia (now Gliwice in Poland) and worked on the waterways that linked the industrial coal-producing area of Silesia to the northern cities. He married Auguste in 1870. The family were Jewish and Ernst was the eldest of his parents three sons: Kurt (born in 1872) was only a year younger than Ernst but Walter (born in 1882) was eleven years younger. Ernst showed a very marked talent for music from a young age and his parents, wishing to give him every opportunity to develop these talents, sent him to the Silesian Music Conservatory where he studied the piano (both playing and composition) for thirteen years [5]:-
He wrote several piano sonatas and, when he was 17, a piano trio.
He prepared for university entrance at the Friedrich Gymnasium in Breslau where his love of mathematics made him decide to study that subject at university rather than music. He entered the University of Breslau in 1890.

After a year at the University of Breslau, Steinitz went to Berlin to study mathematics attending lectures by many famous scientists including Georg Frobenius, Leopold Kronecker and Max Planck. After spending two years in Berlin, he returned to Breslau in 1893. He submitted a solution to a prize problem announced by the university and this won him not only 200 marks but also the right to submit his doctoral dissertation without payment of the usual fee. He worked on this dissertation, advised by Jacob Rosanes, and was awarded his doctorate in 1894 for his thesis Über die Construktion der Configurationen $n_{3}$ . In this work he considers incidence structures in which each line contains exactly 3 points and through each point there are exactly 3 lines. He shows that such a structure could be realised in the Euclidean plane with straight lines except maybe one line which has to be drawn as a quadratic curve. He also proved a result on configurations which in fact is just König's theorem for regular bipartite graphs which Denes König presented to the Congrès de philosophie mathématique in Paris in 1914 and proved in his paper Sur un problème de la théorie générale des ensembles et la théorie des graphes (1923). Steinitz's solution of König's theorem twenty years before König is discussed in detail in [3].

In 1897 Steinitz was appointed Privatdozent at the Technische Hochschule Berlin-Charlottenburg after submitting his habilitation thesis. In the same year he published the paper Über die Unmöglichkeit, gewisse Configurationen $n_{3}$ in einem geschlossenen Zuge zu durchlaufen . He soon came in contact with Issai Schur and Edmund Landau who were both studying in Berlin for their doctorates when Steinitz was appointed. He also became a close friend of Kurt Hensel who had been a Privatdozent at Berlin for ten years before Steinitz arrived. Hensel's grandfather, the Prussian court painter Wilhelm Hensel, had married Fanny Mendelssohn, the eldest sister of the famous composer Felix Mendelssohn. Fanny was herself a famous pianist and composer who wrote about 500 musical compositions in all, including about 120 pieces for piano. Steinitz spent many happy evenings in Hensel's home playing chamber music. Steinitz joined the German Mathematical Society (Deutsche Mathematiker-Vereinigung) in 1897 and there he met Otto Toeplitz. The two became close friends and often played music together.

In 1909 Steinitz was considered for an extraordinary professorship at the University of Würzburg. Although he was not appointed he clearly was considered a strong candidate by David Hilbert who wrote in his support (see for example [5]):-
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.
The offer of a professorship at the Technical College of Breslau saw him return to Breslau in 1910. In the following year, he married his cousin Martha (1875-1942) whose parents were Julius Steinitz (1844-1919) and Rosalie Freund (1847-1937). Steinitz had taught Martha to play the piano when he was at school, continuing the lessons while he was a student. Martha was a very talented pianist who on occasion played duets with the celebrated Austrian pianist and piano teacher Artur Schnabel. Ernst and Martha Steinitz had one child; a son Erhard was born on 6 August 1912. After spending ten years at the Technical College of Breslau, he moved to Kiel in 1920 where he was appointed to the chair of mathematics at the Christian-Albrechts University. This chair, the first in mathematics which was established when the University had been founded in 1665, had become vacant when Heinrich Wilhelm Jung was appointed to Halle to succeed Albert Wangerin. Again Hilbert had supported Steinitz's appointment writing (see for example [5]):-
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.
Below Steinitz he listed Felix Hausdorff and Ludwig Bieberbach. A second chair of mathematics at Kiel had become vacant at about the same time and this was filled by Steinitz's friend Otto Toeplitz who was already an extraordinary professor at Kiel. Steinitz took up his appointment on 30 April 1920 and during his years at Kiel he taught a wide-ranging collection of courses including courses on algebra, polyhedra, number theory, complex analysis, topology, geometry, vector analysis and mechanics. He also conducted a research seminar in collaboration with his colleagues Otto Toeplitz and Helmut Hasse.

Music still played a major role in Steinitz's life. Karsten Johnsen writes in [5] that he:-
... retained his great passion for music throughout his life. There are reports that he could play Beethoven piano sonatas from memory. ... Heinrich Heesch, 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 concerto.
Steinitz died from heart problems in 1928, was cremated on 3 October 1928 in Lübeck but his ashes were buried in Breslau. Only a few weeks before his death Erich Hecke wrote to Adolf Kneser, who was at Breslau, 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.
After the death of her husband, Martha returned to Breslau with their sixteen year old son Erhard but after the Nazis came to power in 1933 Martha and Erhard went to Palestine. Given the musical talents of his parents, it will come as no surprise to learn that Erhard was very musical. In fact he played bassoon in the Israel Philharmonic when it was first formed. Sadly, Martha chose to return to Breslau where she became a victim of the Nazis. She was sent to Theresienstadt, a transit concentration camp, before being sent to the Treblinka concentration camp where she died in the gas chambers in the second half of 1942. Sadly Erhard Steinitz, who married Ilse Schlesinger, died in 1948 at the age of 36.

The direction of Steinitz's mathematics was much influenced by Heinrich Weber and by Hensel's results on $p$-adic numbers in 1899. In [4] interesting results by Steinitz are discussed. These results were given by Steinitz in 1900, when he was a Privatdozent at the Technische Hochschule Berlin-Charlottenburg, at the annual meeting of the Deutsche Mathematiker-Vereinigung in Aachen. In his talk Steinitz introduced an algebra over the ring of integers whose base elements are isomorphism classes of finite abelian groups. Today this is known as the Hall algebra. Steinitz made a number of conjectures which were later proved by Hall.

Steinitz is most famous for work which he published in 1910. He gave the first abstract definition of a field in Algebraische Theorie der Körper in that year. Prime fields, separable elements and the degree of transcendence of an extension field are all introduced in this 1910 paper. He proved that every field has an algebraically closed extension field, perhaps his most important single theorem. The now standard construction of the rationals as equivalence classes of pairs of integers under the equivalence relation: $(a, b)$ is equivalent to $(c, d)$ if and only if $ad = bc$ was also given by Steinitz in 1910. In 1930, twenty years after its first publication, Steinitz's paper Algebraische Theorie der Körper was republished with a foreword by Reinhold Baer and Helmut Hasse. They wrote:-
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
Steinitz also worked on polyhedra and, after his death, his manuscript on the topic was edited by Hans Rademacher and published as Vorlesungen über die Theorie der Polyeder in 1934. Donald Coxeter writes [2]:-
Most people who write about polyhedra are concerned with special forms, such as regular or crystallographic polyhedra. Steinitz has deliberately kept away from special figures, save as examples, since they have been fully treated in such works as Brückner's 'Vielecke und Vielflache'. Many results which one is tempted to take for granted are here proved, with admirable thoroughness and rigour. Subsequent authors will surely be grateful that they can quote such results, instead of either assuming them as "obvious" or trying to prove them for themselves. The general theory of polyhedra - even of convex polyhedra - is rather a tantalizing subject. The natural question, "How many types of polyhedra have $n$ vertices?", can be answered for small values of $n$, and anyone who is sufficiently industrious can take values a little higher than his predecessors; but very little is known when $n$ is arbitrary. Steinitz introduces polyhedra from many different aspects: metrical, combinatorial, topological, analytical, projective and axiomatic. Then, having laid the foundations, he is content to stop. This treatment causes a certain lack of unity in structure, and the reader must not expect to find any startling conclusions. Each general statement is amply illustrated by examples; this accounts for the book's great length, but makes it more readable than it could otherwise be. Much of the latter part is concerned with proving in various ways the 'Fundamental Theorem of Convex Types'. Steinitz defines a "K-polyhedron" as a Eulerian polyhedron with the property that whenever two faces have two vertices in common, the join of these vertices is an edge. Clearly every convex polyhedron is a K-polyhedron. The "Fundamental Theorem" states that every K-polyhedron is isomorphic with a convex polyhedron. Although the book has only a paper cover, it is elegantly printed on good paper, and the 190 diagrams leave nothing to be desired.
Finally let us mention some of Steinitz's other mathematical contributions. He published important papers on module theory, for example Zur Theorie der Moduln (1899) and the two-part paper Rechteckige Systeme und Moduln in algebraischen Zahlköppern (1911, 1912). He also wrote several significant papers on conditionally convergent series such as Bedingt konvergente Reihen und konvexe Systeme (1913). In fact Edmund Landau had suggested to Steinitz that he read Paul Lévy's 1905 paper on conditionally convergent series but Steinitz found this unsatisfactory writing in his 1913 paper that Landau's suggestion:-
... prompted me to study [Lévy's] 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 M 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 M 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. ...
In fact, perhaps surprisingly, it was in the context of his work on series that he set out what today is often called the 'Steinitz replacement theorem' for vector spaces. In Bedingt konvergente Reihen und konvexe Systeme (1913) he stated and proved the replacement theorem in the following form (we have changed Steinitz's language into modern terminology):-
Let $V$ be a vector space which is spanned by $n$ elements. If $V$ contains $r$ linearly independent elements $v_{1}, v_{2}, ..., v_{r}$, there is a spanning set of $V$ consisting of $n$ elements and containing the $v_{i}, 1 ≤ i ≤ r$. In particular, a vector space spanned by $n$ elements cannot contain more than $n$ linearly independent elements.

### References (show)

1. B Schoeneberg, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.
2. H S M Coxeter, Review: Vorlesungen über die Theorie der Polyeder by E Steinitz, edited by H Rademacher, The Mathematical Gazette 19 (232) (1935), 56-57.
3. H Gropp, On combinatorial papers of König and Steinitz, Algebra and combinatorics: interactions and applications, Königstein, 1994, Acta. Appl. Math. 52 (1-3) (1958), 271-276.
4. K Johnsen, On a forgotten note by Ernst Steinitz on the theory of abelian groups, Bull. London Math. Soc. 14 (4) (1982), 353-355.
5. K Johnsen, Famous scholars from Kiel: Ernst Steinitz. Mathematician, author of seminal works on algebra, Professor of Mathematics in Kiel from 1920 to 1928. http://www.uni-kiel.de/ps/cgi-bin/fo-bio.php?nid=steinitz&lang=e