At this time Frankfurt University had some extremely talented mathematics lecturers who could inspire their students. Max Dehn (appointed in 1921) held the chair of Pure and Applied Mathematics, Paul Epstein (appointed in 1919) was a professor, Ernst Hellinger (appointed in 1914) held a chair, Otto Szász (appointed in 1914) was a professor, and Carl Siegel (appointed in 1922) held a chair. Frankfurt University, with this array of teaching talent, was an exciting place when Schneider entered it, and as a consequence the number of students taking mathematics courses had built up over the previous few years. From a position of almost no students in 1920, by 1928 there were 143 students taking the first semester differential and integral calculus course. A student as mathematically talented as Schneider was always going to be converted to mathematics by such inspiring teachers, and indeed it was not long before he had dropped other sciences to concentrate on mathematics. A course on transcendental numbers, given by Carl Siegel, was so inspiring that Schneider decided that he wanted to enter Siegel's research seminar and he passed the difficult entrance examination. Siegel gave Schneider a number of possible problems which he might work on for his doctorate. However, Schneider had so loved Siegel's course on transcendental numbers that he began to look at one of the open problems that Siegel had listed in that course.
David Hilbert's Seventh Problem, given in his address to the Paris International Congress of Mathematicians in 1900, asks:
Prove that ab is transcendental where a is algebraic (not 0 or 1) and b is an algebraic irrational number.Aleksandr Osipovich Gelfond proved a special case of Hilbert's Seventh Problem in 1929 when he showed that the result held for a algebraic (not 0 or 1) and b an imaginary quadratic. Using Gelfond's ideas, Siegel showed how to prove this for a real quadratic b and he gave an indication of this in his course on transcendental numbers that Schneider attended. In fact Siegel did not publish his result and Rodion Osievich Kuz'min also saw how to use Gelfond's ideas to prove the Seventh Problem in the case where b is a real quadratic and published in 1930. Siegel had indicated in his course that the general problem was still open, although he did not explain that this was Hilbert's Seventh Problem. Schneider later wrote (see ):-
After a few months, I gave Siegel six pages of my work, and was then told by him that I had solved Hilbert's Seventh Problem.In fact Gelfond had also, independently, managed to extend the ideas in his 1929 paper to complete the proof of Hilbert's Seventh Problem so the result is now known as the Gelfond-Schneider Theorem. Schneider published his proof of Hilbert's Seventh Problem in the paper Transzendenzuntersuchungen periodischer Funktionen Ⓣ (1934) which appeared in Crelle's Journal. [Gelfond's proof was published in Sur le septième Problème de D Hilbert Ⓣ (1934).] One might imagine that solving one of Hilbert's Problems would be enough for a Ph.D. thesis but Siegel thought that something so short (five pages in the published paper) might not be acceptable to the Frankfurt Faculty. Schneider then produced another five pages on transcendence of elliptic functions and the two parts formed his thesis, also entitled Transzendenzuntersuchungen periodischer Funktionen Ⓣ, which earned him a doctorate in 1934.
One might think that solving one of Hilbert's Problems in your Ph.D. thesis would make getting a job easy, but Germany in 1934 was not operating by the usual rules. Enthusiastic support for the Nazi Party and its ideology was far more important that academic ability :-
... from the autumn of 1933, success did not depend any longer on academic achievement, but rather on being a member of a suitable Nazi organisation and having views consistent with Nazi ideology. If Schneider wanted to carry on with his studies or find a university post, he would have to join a Nazi organisation. The alternative would be to give up any idea of an academic career.Schneider was not Jewish but he was closely associated with his teachers Dehn, Epstein, Hellinger and Szász who were all Jewish. His closest association was with Carl Siegel who, although not Jewish, made no secret of his anti-Nazi views and this was perhaps the most damaging of all Schneider's difficulties with the Nazis. He had to make the choice, as indicated in the above quote, and he chose to join the Sturmabteilung (SA), the paramilitary organisation of the Nazi Party. This might look at first as if Schneider was going too far in his efforts to be accepted, but in fact this is not so. Although the SA had played a major role in Hitler's rise to power, the organisation became essentially irrelevant after June 1934 when Hitler, believing that the SA leaders were plotting against him, had over 100 of the SA leaders arrested and shot. The SS then became the Nazi Party's paramilitary wing and the SA became largely irrelevant. Schneider's move worked and he was appointed as an assistant at Frankfurt am Main in 1935. However, it was still known by the authorities that he was not a supporter of Nazi ideology and, for that reason, he was not allowed to attend the International Congress of Mathematicians in Oslo in July 1936. Had he been allowed to attend, he would have presented his proof of Hilbert's Seventh Problem.
The work on transcendence which had led Schneider to his solution of Hilbert's Seventh Problem led him to extend to a more general programme studying the transcendence of elliptic functions, modular functions and abelian functions. He published three papers in 1936, one being Arithmetische Untersuchungen elliptischer Integrale Ⓣ and, two years later, he submitted his habilitation thesis Zur Theorie der Abelschen Funktionen und Integrale Ⓣ to Frankfurt University. It was a remarkable and important piece of work but the thesis was rejected. Although we have no evidence that it was never read, many (including Yandell ) believe that it was rejected without ever being given any consideration. With Nazi policies becoming ever more extreme, Siegel had found the situation in Frankfurt intolerable; Szász had been dismissed and Dehn, Epstein and Hellinger, although still living in Frankfurt, were not allowed to teach at the University. In late 1937 Siegel accepted a professorship in Göttingen and moved there in 1938. After his habilitation was rejected by Frankfurt, Schneider moved to Göttingen to become Siegel's assistant (a very lowly position for someone with his outstanding record). On 9 November 1939 he submitted his habilitation thesis Zur Theorie der Abelschen Funktionen und Integrale Ⓣ to Göttingen. It was identical to the thesis which had been rejected by Frankfurt one year earlier, but it was accepted by Göttingen and in 1940 he was appointed to a teaching position at Göttingen.
Of course, by 1940 Germany was at war and Schneider was drafted into the German army. He undertook war service as a meteorologist, mostly stationed in France. During this period, his habilitation thesis was published in Crelle's Journal in 1941. However, this was his only publication between 1936 and 1949, the years that, under normal circumstances, would have almost certainly been the most productive of his life. In 1944 Schneider, still undertaking war service, received a request to go to the University of Göttingen to cover for Helene Braun who had contracted diphtheria. This request came due to the efforts of Wilhelm Süss who was setting up a Mathematical Institute at Oberwolfach. The Nazis believed that the Oberwolfach Mathematical Institute was being set up to help the war effort, but Süss was attempting to save as many German mathematicians from dying as he possibly could in the final days of the war. Once Schneider was in Göttingen, he was brought by Süss to Oberwolfach in March 1945. Along with about 20 other mathematicians, he remained there until the war ended. Süss's wife, Irmgard, wrote about this time and she:-
... tells a tale of courage and camaraderie in those final days; the problem of securing food and heating; the preparations for flight in case the Institute was attacked; the eventual occupation of the lodge; and, once the war was officially over, the frantic burning of books on National Socialism that had been stored in the house.In May 1945 Germany surrendered but Schneider remained at the Oberwolfach Mathematical Institute until the autumn of that year by which time Göttingen University had reopened. Göttingen is nearly 500 km by road from Oberwolfach but Schneider made the journey :-
... on a rattletrap bicycle put together from junked parts.He was back to an assistant position at Göttingen where he remained until 1953 except for the academic year 1947-48 which he spent at the University of Münster as a substitute professor. In 1950 he married Maria Urbach :-
... finally, unqualified good fortune. He called her "Mieke" and, while he is described as calm, she was full of energy and expression. Schneider's students, writing about her, made it clear how much they appreciated her. Schneider was thirty-nine and Mieke thirty-five when they married. They had one child, Bernard, who became a doctor.In 1953 Schneider was named as an ordinary professor at Erlangen. He remained there until 1959, being head of the Faculty of Science during 1955-57. In 1959 he went to Freiburg to fill the chair left vacant following the death of Süss from liver cancer in May 1958. Süss had been the first director of the Oberwolfach Institute but, following his death, Hellmuth Kneser served as director of the institute until Schneider was appointed to that role in 1959. He held this position for four years until Martin Barner took over in 1963. Schneider kept his close association with Oberwolfach, however, being an organiser of the number theory meetings (along with Helmut Hasse and Peter Roquette) held every year or two from 1955 to 1972. After 1972 he organised Oberwolfach meetings on diophantine approximation and transcendental numbers.
He retired in 1976, at the earliest opportunity, but remained in his home in Zähringen, Freiburg :-
... enjoying, with his wife, the garden, his sports car, vacation trips, and freedom from all obligations.Let us look briefly at the important monograph Einführung in die transzendenten Zahlen Ⓣ (1957) by Schneider which was also published in French translation two years later as Introduction aux nombres transcendants Ⓣ. Kurt Mahler, who himself made important contributions to transcendence of numbers, writes in a review:-
This addition to the small library of modern books on transcendental numbers ... will be welcomed by mathematicians everywhere, the more so since it contains proofs of the author's fundamental work on the transcendency of elliptic and modular functions. (His even more far-reaching results on abelian functions are also mentioned, but not proved.)Schneider remained close friends with Siegel, often enjoying walks together in the country in their retirement, and was greatly affected by his death in April 1981. However, Schneider's wife and family gave him the happiness in his final years which had eluded him through the troubled years when he was trying to carve out a career with the odds stacked firmly against him.
Article by: J J O'Connor and E F Robertson