In 1923, while at the Kaiserin Augusta Gymnasium, Scholz was awarded his Abitur, giving him the right to study at university. He entered the University of Berlin in the summer semester of 1923 where he studied mathematics, philosophy and musicology. Many German students of this time took courses at a number of different German universities but Scholz chose to spend only one semester away from Berlin and that was at the University of Vienna in the summer of 1927. By this time in his career, Scholz was undertaking research for his doctorate advised by Issai Schur and his research had taken him into areas to which Philipp Furtwängler had made substantial contributions. In Vienna he worked with Furtwängler who had, by that time, been a professor at Vienna for over ten years. Furtwängler was severely disabled by this stage in his tenure of the chair of mathematics and was confined to a wheelchair. One of the students who was undertaking research in Vienna advised by Furtwängler was Olga Taussky (later Taussky-Todd). She writes :-
While I was working on my thesis, a young German, a pupil of Schur, turned up. He was working on his thesis, which was related to Furtwängler's ideas. I saw him talk to Furtwängler. I gathered that he was very gifted and interested in computing.Scholz was looking at the inverse Galois group problem, namely whether a given finite group can be the Galois group of some extension of the rational numbers. The problem in general is still unsolved but Scholz was solving special cases.
Even before going to Vienna, Scholz had read material produced by Helmut Hasse on class field theory. He wrote to Hasse on 22 April 1927, on the day before travelling to Vienna, suggesting ways that proofs of Hasse's results could be substantially simplified and, as a result, Scholz and Hasse wrote the joint paper Zur Klassenkörpertheorie auf Takagischer Grundlage Ⓣ which was published in Mathematische Zeitschrift in 1928. It became Scholz's first paper.
After his semester in Vienna, Scholz returned to Berlin in September 1927 and continued work on his thesis Über die Bildung algebraischer Zahlkörper mit auflösbarer Galoisscher Gruppe Ⓣ. On 1 April 1928 he was appointed as an assistant to Erhard Schmidt. In October 1928 Scholz wrote to Hasse saying that he had constructed number fields with arbitrarily large class field towers. This was an important result and Hasse immediately informed Emil Artin. Scholz published this result in his paper Zwei Bemerkungen zum Klassenkörperturm Ⓣ which appeared in Crelle's Journal in 1929. He submitted his doctoral dissertation and was examined by Issai Schur and Erhard Schmidt on 19 December 1928. He was awarded his doctorate, magna cum laude, in the following year. Scholz adds the same words to both his thesis and to the paper consisting of his thesis material published in Mathematische Zeitschrift in 1929:-
I would like to express my special thanks to Professor I Schur and Professor Ph Furtwängler for their friendly advice on the preparation of this work.This paper consisting of his thesis material was one of four papers by Scholz published in 1929. The other three are: Reduktion der Konstruktion von Körpern mit zweistfiger (metabelscher) Gruppe Ⓣ; Anwendung der Klassenkörpertheorie auf die Konstruktion von Körpern mit vorgeschriebener Gruppe Ⓣ; and Zwei Bemerkungen zum Klassenkörperturm Ⓣ mentioned above.
In April 1929, after the award of his doctorate, Scholz was appointed as an assistant to Alfred Loewy at the University of Freiburg. This position had become vacant since Reinhold Baer, who had been Loewy's assistant at Freiburg from 1926, had moved to Halle in 1928. Scholz habilitated at the University of Freiburg. Ernst Zermelo had held a chair in Zurich but had been forced through ill health to resign this chair in 1926. However, he had been appointed to an honorary chair in Freiburg and so became an important colleague for Scholz. On 8 July 1929 Scholz wrote (see for example ):-
I have much more free time in Freiburg ... I just have to take care of Loewy's seminar. I am also receiving sufficient suggestions from Loewy to look at various papers. Zermelo, however, is much more interesting to me. After having got through Warsaw in good shape, he is currently in Zoppot. He will return here in August.In fact Scholz's interest in Zermelo's ideas is clear from the fact that he published Zermelos neue Theorie der Mengenbereiche Ⓣ in 1931.
We mentioned above that Olga Taussky had met Scholz when he spent the summer of 1927 in Vienna. She met him again in 1930 and they began a joint collaboration at that time. Olga Taussky writes :-
In 1930 I went to Königsberg where a meeting of the Deutsche Mathematiker-Vereinigung took place (Hilbert gave a famous lecture on logic there). I gave a lecture on my thesis and met famous people like Noether and Hasse. I recognised Scholz and since our mathematical interests were clearly related (he was a few years older and had already several publications by then) we started a conversation. Soon after I returned to Vienna I had a letter from him suggesting joint work by correspondence ...Another professor at Freiburg when Scholz began working there was Lothar Wilhelm Julius Heffter (1862-1962). Heffter, who had been a student of Lazarus Fuchs, had been a professor in Freiburg since 1911. He retired from his chair in 1931 but continued to live in Freiburg. In the summer of 1931 Gustav Doetsch was appointed to succeed Heffter and, at this time, Scholz changed from being Loewy's assistant to become an assistant to Gustav Doetsch. However, it was Zermelo with whom Scholz formed a close friendship and he tried hard to arrange a meeting between Zermelo and Kurt Gödel at the Deutsche Mathematiker-Vereinigung meeting held in Bad Elster in September 1931. Olga Taussky, who was at the same meeting, wrote :-
I worked then with the number theoretician Arnold Scholz, who was a great class field expert. Both Scholz and Zermelo worked in Freiburg. Scholz was eager to help Zermelo and thought a discussion with Gödel would achieve this. ... The trouble with Zermelo was that he felt he had already achieved Gödel's most admired result himself. Scholz seemed to think that this was in fact the case, but he had not announced it and perhaps would never have done so. ...In the spring of 1932 Scholz, together with Zermelo, went on the "Hellas Tour" for teachers and students from German High Schools. The cruise with the steamship "Oceana" was from 19 April to 4 May and visited mainly places in Greece, but also places of interest in Italy and North Africa. In early 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. At first it might look as if the Aryan Scholz would be unaffected by this but in fact this was far from the case and over the next few years Scholz's life was made extremely difficult because of the Nazi agenda. The immediate problem in April 1933 was that Loewy, who was a Jew, was dismissed and this left Doetsch as the only full professor at Freiburg. Doetsch gave his wholehearted support to the ideas of National Socialism as put forward by the Nazis and declared that algebra and number theory had become, like all areas under Jewish influence, completely abstract and un-German. Within three months of the Nazis coming to power, Scholz was trying to get away from Freiburg.
In the summer of 1933, Scholz visited Berlin and while there he talked with Erhard Schmidt about the possibility of transferring his habilitation to Berlin. Later that year he applied for a grant from the Emergency Association. Both Heffter and Doetsch were asked their opinion of Scholz's work by those deciding whether to award him a grant from the Emergency Association and they replied at the end of April and the beginning of May, rspectively. In 1934 Scholz was awarded the grant and, resigning as Doetsch's assistant, he continued to search for another position. He explained his reasons in a letter to Olga Taussky, saying that Wilhelm Süss had been appointed to fill the chair that had become vacant when Loewy was dismissed. Scholz has been told by Georg Feigl that Süss was a very nice person but he was not an algebraist. Meanwhile Doetsch had essentially eliminated higher algebra lectures from the Freiburg syllabus so Scholz now felt superfluous.
Scholz next wrote to Hasse asking if he could transfer his habilitation to Göttingen. Hasse had been appointed to Göttingen after Hermann Weyl, who had a Jewish wife, had resigned his chair there. Hasse, however, told Scholz he had no chance of a position at Göttingen. Friedrich Karl Schmidt (1901-1977) suggested to Scholz that he might be able to get an appointment in Kiel. On 31 July 1934, Doetsch wrote a letter to Erhard Tonier about Scholz which illustrates the extremely difficult situation that Scholz was in at Freiburg and shows why he was so keen to move (see  or ):-
I can only say one thing, that in my experience such a lack of a sense of duty at German universities has not yet occurred. ... I do not ask anyone whether he is a National-Socialist, I'm not even a party member. I would, in line with the intentions of the government, promote only someone who at least has a positive attitude to the present state. There can be no question about Scholz. He is the opposite, known here together with his only friend he has, namely Zermelo, as the typical complainers who saw nothing better than that the National Socialist Government disappears as soon as possible because, in contrast to previous governments, it threatens to pursue such characters as him. It is, however, out of the question that such a softy as Scholz would be able to exert a political influence. ... But if people like Scholz see that today, despite all government announcements, they are doing very well, [Footnote: Six months ago, he received a grant for himself from the Emergency Committee; the matter was initiated behind my back] nobody should not be surprised if they continue their mischief calmly and are even more insolent. ... You will now perhaps understand that I was utterly amazed that this man is to be awarded a teaching position, just at a time when the fiercest battle has been announced against such drones. ... Students are opposed to him, completely dismissive, in the previous semester they even wanted to take action against him, probably mainly because he went around exclusively with Jews and Communists.In 1935 Scholz's grant from the Emergency Association ended and in August he made his first visit to Kiel. He settled there in October, but doubled his efforts to get a permanent appointment. However his refusal to conform to what the Nazis wanted made this exceptionally hard. Reports on him stated that (see for example ):-
Politically Scholz is not reliable. ... at Freiburg he was permanently together with Jews and former communists. Scholz will not understand Nazism.In the summer of 1935 Scholz had applied for a position as an assistant to Friedrich Karl Schmidt at Jena. Negotiations went on until October but, at that time, he received a letter from the rector of Jena saying that the Reich Ministry of Culture wanted him to remain in Kiel. F K Schmidt wrote to Hasse in late October regretting that he was not able to employ Scholtz but, since the Reich Ministry of Culture had said that he should stay in Kiel, Schmidt expected Scholtz to get a teaching position in Kiel. However, by December Schmidt was writing again to Hasse saying (see for example ):-
Much to my surprise I heard a few days ago from Scholz that he has still not received the teaching position in Kiel. ... In any event as long as it is still a possibility for Scholz to get something in Kiel, I think it would be unwise if he leaves his post and takes on a foreign appointment.Scholz cerytainly considered trying to get a position abroad but he did not want to leave Germany and he was being advised to remain in Kiel. He did consider a number of possibilities such as Amsterdam, Cambridge Massachusetts, and Yugoslavia. He also asked Hasse if Oystein Ore could find a way to support him at Yale. Ore replied to Hasse in January 1936 (see for example ):-
I met Scholz in Hamburg last year and he made the best impression upon me. I am afraid I shall have to be rather discouraging about the possibilities for obtaining a position in America at the present time, but there might still be some chances. ... I should, however, not be too hopeful on the outcome of an application. The Institute of Advanced Study in Princeton announces a set of fellowships in the last number of the Bulletin of the American Mathematical Society. Scholz may also apply for one of these ... .Scholz continued to seek positions, for example in Tübingen. However, in addition to a continual stream of high quality research papes, he published the little book Einführung in die Zahlentheorie Ⓣ in 1939. This was reprinted in 1945 without changes, and in 1955 an edited version was published. Reviewing the 1939 edition Louis Mordell writes :-
This is an introduction in the more proper sense of the word. The author gives a strictly logical development of the elements of the classical theory, using, however, modern ideas and modes of expression when they are appropriate. He also, when it fits in with his scheme, gives some more advanced results not usually found in elementary books but none the less most welcome in enlarging the reader's horizon. ... It is quite clear that the author has produced an excellent little book. It contains a great deal in a small compass but does not suffer in any way from condensation. It can be strongly recommended to all desirous of a cheap, convenient and rapid introduction into number theory which takes note of the influence of modern algebra upon the subject.Reviewing the 1945 reprint, Harold Davenport writes:-
The book gives, in a small compass, a good account of classical number-theory, with occasional inclusion of more recent results. It consists of six chapters: (1) the arithmetic of the natural numbers, (2) divisibility properties, (3) congruences, (4) quadratic residues, (5) quadratic forms, (6) algorithmic calculation. ... The author pays greater attention to logical structure than is usual in elementary books, and also stresses the general concepts of group, ring and field at appropriate points. The exposition is somewhat condensed (e.g., Fermat's proof of the insolubility of x4 + y4 = z2 takes only 8 lines), but is always adequate. The book may be strongly recommended to anyone who seeks a concise account of elementary number-theory.Reviewing the 1955 edition, which had been revised by B Schoeneberg, Albrecht Fröhlich writes :-
This little book should prove to be an attractive and stimulating introduction to Number Theory. It cannot attempt to touch upon all the main branches of the subject, but instead concentrates on a few principal topics and thus succeeds in leading the reader up to interesting fundamental results of a comparatively advanced nature.While in Kiel Scholz supervised the doctoral studies of Gunter Hannink. However, in May 1940 Scholz was conscripted into the army and sent as a radio operator to the East. In 1941 he was assigned to the naval college in Flensburg-Mürwik as a teacher of mathematics. The naval station and academy at Flensburg had been established before World War II, and served an important role for Germany throughout the war. At Flensburg, Scholz was a colleague of Ott-Heinrich Keller. In December 1941 he was told that he might not be allowed to return to Kiel after the war ended. However, this quickly became irrelevant since Scholz died on 1 February 1942 in Flensburg. It is unclear exactly what caused his death but a form of pulmonary tuberculosis seems the most likely.
Although Scholz published results of major importance in the top journals, his work seems not to have been influential except perhaps in as far as it influenced Hasse. The reason for the lack of influence may be that Scholz was one of those brilliant mathematicians who are so far ahead of their time that they fail to make an impact since contemporaries fail to understand the significance of their results. For example the editors of  write:-
It is not an exaggeration to say that in the 1930s, Arnold Scholz belonged to the elite of the younger algebraic number theorists in Germany. In particular, he worked on the generalisation of Hasse's norm theorem to arbitrary normal extensions. ... It seems, however, that few people except Hasse had any idea about the depth of Scholz's work on the norm theorem because his article is, to say the very least, very difficult to read. It was Jehne [Wolfram Jehne, On knots in algebraic number theory. In memoriam Arnold Scholz (1979)] who realised that Scholtz had basically formulated Tate's result [John Tate, Global class field theory (1967)] on the obstruction to Hasse's norm theorem in non-cohomological terms and remarked that "Scholz's investigations have been forgotten for almost 40 years".
Article by: J J O'Connor and E F Robertson
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