In 1900 he went to Göttingen University where he attended lectures by Hilbert and Klein. At this time Klein was running a seminar which studied the theory of elasticity, descriptive geometry, and mechanics, and Hamel participated in this seminar. His doctoral work was supervised by Hilbert and in 1901 he was awarded a doctorate for his thesis Über die Geometrien, in denen die Geraden die Kürzesten sind which studied the Fourth Hilbert Problem. Also in 1901 he passed the state examinations to enable him to teach mathematics and physics in secondary schools. He was appointed as an assistant to Klein during the year 1901-02. He was then appointed assistant to Karl Heun at the Technical University of Karlsruhe in the autumn of 1902 and he presented his habilitation dissertation to the Technical University of Karlsruhe one year later, qualifying to lecture in mathematics and mechanics.
The German Technical University of Brünn was looking for a professor to fill the Chair of Mechanics which fell vacant when Karl Hellmer retired. After rejecting several Austrian candidates as unsuitable, a committee of the University drew up a list of four possible candidates. They placed Hamel in first place stating that :-
He is the youngest of all the candidates but with his knowledge and scientific achievement, he would be a credit to any university.Hamel was appointed Professor of Mechanics at the German Technical University of Brünn on 3 October 1905. He was one of four mathematicians considered to fill Stäckel's chair at the Technical University of Hanover in March 1908. The criteria for that post were that:-
... the professor to be appointed, besides having fully proved teaching skills and a comprehensive knowledge of mathematics, should also have a clear understanding of its applications to engineering.Klein spoke highly of Hamel in his reference but considered Carathéodory superior for the particular position, and Carathéodory was appointed. In August 1909 Hamel married Agnes Frangenheim in Cologne. They had three daughters. Hamel was appointed to the chair of mechanics at the Rheinisch-Westfälische Hochschule in Aachen on 1 October 1912, then in 1919 he moved to the Technical University of Charlottenburg in Berlin where he was appointed as professor of mathematics and mechanics.
On 30 January 1933 the National Socialist party led by Hitler came to power in Germany and Nazi policies began to have a major impact on the German Mathematical Society and its members. Hamel was a leading member of the German Mathematical Society and was much involved with the political moves which went on in that Society. In fact he had been chairmen of the Reich Mathematics Association since its foundation in 1921. This Association was a branch of the German Mathematical Society and provided in some sense a political wing. Hamel was clearly associated with the views of National Socialism and in 1933 spoke of a spiritual bond between mathematics and the "Third Reich".
In 1934 Bieberbach made a bid to become chairman of the Society but his bid failed and Blaschke was elected Chairman of the German Mathematical Society in September 1934. However Bieberbach, as secretary to the Society, managed to prevent changes to the statutes of the Society that Blaschke attempted to introduce and the Reich Ministry of Education intervened forcing both Bieberbach and Blaschke to resign. Blaschke resigned the Chairmanship of the German Mathematical Society in January 1935 and Hamel was appointed chairman to replace him. Although Hamel met with approval by the National Socialists, his leadership was based on mathematical scholarship despite his close links with Nazi policies. He undertook editorial duties beginning in the spring of 1935 for the Neue Deutsche Forschungen, a new journal which was set up when the Germans withdrew their support from Compositio Mathematica on the grounds that it was publishing too many works by non-Aryans. Together with Süss and Behnke, Hamel also served on an Instruction Commission set up by the German Mathematical Society. After World War II ended Hamel was appointed to the Technical University of Berlin in 1946. He was a visiting professor at the University of Tübingen during 1946-47.
Hamel worked in function theory, mechanics and the foundations of mathematics. He is perhaps best known for the Hamel basis, published in 1905, when he made an early and explicit use of the Axiom of Choice to construct a basis for the real numbers as a vector space over the rational numbers. He wrote a number of papers on an axiomatic theory of mechanics, the first two in 1909. His first textbook on mechanics was Elementare Mechanik which was published in 1912 :-
This work, which was more than 600 pages long, was devoted not only to the elements of mechanics, but also to a number of its special fields. it was based on his lectures at the Technical University [of Brünn] dealing with the elements of statics, dynamics, and hydrodynamics, as well as more advanced problems.He wrote papers on differential equations, and on fluid dynamics, for example on the critical velocity of a fluid, that is the velocity at which the flow changes from laminar to turbulent. He made other interesting contributions such as his work on the mechanical "ciphering machine" invented by the engineer Alexander von Kryha from Berlin. In 1927 Hamel computed the size of the key space of the Kryha-Ciphering-Machine, which was quoted extensively by "Internationale Kryha-Maschinen-Gesellschaft" (Hamburg) to infer the unbreakability of the Kryha machines.
Article by: J J O'Connor and E F Robertson