Furtwängler attended elementary school in his home town of Elze and then he studied at the Andreanum Gymnasium in Hildesheim. This Gymnasium was the nearest to Elze, with Hildesheim being only about 12 km distant. He graduated from the Gymnasium at Easter 1889 and, later that year, he began his study of mathematics at Georg-August University of Göttingen. This was perhaps the best university in the world for mathematics at this time and he was taught by a number of leading mathematicians; Heinrich Bürkhardt, Robert Fricke, Otto Hölder, Felix Klein, Arthur Schönflies and Wilhelm Weber. In addition to mathematics, which formed his main interest, he also studied courses in physics and chemistry. Furtwängler attended Felix Klein's number theory course which went on for two semesters. This course particularly interested him and Klein suggested that he might undertake research on ternary cubic forms for his doctorate. Furtwängler wrote the thesis Zur Theorie der in Linearfaktoren zerlegbaren ganzzahlingen ternären kubischen Formen and he was awarded his doctorate by the University of Göttingen for this work in 1895. This thesis was published as a 64-page monograph in 1896. This was not his first publication for he had published Zur Begründung der Idealtheorie in 1895. In fact, he had completed his studies at Göttingen in 1894 and he spent the academic year 1894-95 as an assistant at the Physics Institute of the Technical University of Darmstadt.
At the University of Göttingen, Furtwängler had qualified to teach mathematics, physics, chemistry and mineralogy at secondary schools. However, before beginning secondary school teaching, he spent a year doing voluntary military service in a programme specially designed for those who had undertaken higher education. After this year, he spent two years teaching in schools in Hanover, Norden, a town on the North Sea shore, and Celle which is situated in north-central Germany. The city of Celle is on the Aller River, at the southern edge of the Lüneburger Heide, northeast of Hanover. After this necessary teaching experience to complete his teaching qualifications, in 1898 he was appointed as an assistant at the Geodetic Institute in Potsdam. He was promoted to research associate and remained there until 1904. In this post he carried out, in collaboration with Friedrich Kühnen, measurements of absolute masses and measurements of the gravitational constant at various locations in Silesia. He contributed the article Die Mechanik der einfachsten physikalischen Apparate in Versuchsanordnungen to the Encyklopädie der mathematischen Wissenschaften which was published in 1904. While undertaking this work, he married Ella Buchwald in 1903; they had one daughter.
Despite working at the Geodetic Institute in Potsdam, it was number theory which was Furtwängler's main research interest. His 80-page paper Über das Reziprozitätsgesetz der l-ten Potenzreste in alqebraischen Zahlkörpern, wenn l eine ungerade Primzahl bedeutet was awarded a prize by the Göttingen Academy of Sciences. This was a contribution to Hilbert's Ninth Problem, stated in his address at the International Congress of Mathematicians in Paris in 1900. David Hilbert presented the following form of the Ninth Problem in his paper in the conference Proceedings:
Proof of the most general reciprocity law.Furtwängler continued to attack Hilbert's Ninth Problem in a series of papers: Allgemeiner Existenzbeweis für den Klassenkörper eines beliebigen algebraischen Zahlkörpers (1907); Reziprozitätsgesetze für Potenzreste mit Primzahlexponenten in algebraischen Zahlkörpern I , (1909); Reziprozitätsgesetze für Potenzreste mit Primzahlexponenten in algebraischen Zahlkörpern II (1912); and Reziprozitätsgesetze für Potenzreste mit Primzahlexponenten in algebraischen Zahlkörpern III , (1913).
For a number field, it is to be proved the most general reciprocity law for the p th power residues, when p denotes an odd prime, and moreover, when p is a power of 2 or a power of an odd prime. The law, as well as the means essential to its proof, will, I believe, result from suitably generalizing the theory of the field of the p th roots of unity, developed by me, and any theory of relative quadratic fields.
Let us return to our description of Furtwängler's career. He had qualified as a secondary school teacher and then worked at the Geodetic Institute in Potsdam but he had never habilitated. However, he was appointed as a professor at the Agricultural Academy in Bonn in 1904 and worked there for three years before moving to Aachen where he was a professor of mathematics at the Technische Hochschule from 1907 to 1910. He then returned to the Agricultural Academy in Bonn as a professor of mathematics where he taught during 1910-1912 while at the same time held a lectureship in applied mathematics at the University of Bonn. In 1912 he was appointed as a professor of mathematics at the University of Vienna and he continued to hold this post until he retired in 1938. In this role he worked very hard. Furtwängler came to the Mathematical Institute at 8 clock in the morning and held oral examinations and colloquia. From 10 o'clock he held a one-hour seminar twice weekly. He regularly delivered lectures on differential and integral calculus, on number theory and on algebra. These lectures were very well attended; more than four hundred students attended his lectures but only half of them could get seats in the lecture theatre. This was all the more remarkable since, from 1916 he became progressively more paralysed and lectured from a wheelchair. He lectured without notes and had an assistant who wrote equations on the blackboard. It was Furtwängler's lectures which turned Kurt Gödel from physics to mathematics. Gödel, who entered the University of Vienna in 1922, described Furtwängler's lectures as the best he had ever heard. To hear this quality of lectures from someone confined to a wheelchair who was paralysed from the neck down, would make a big impact on any student, but on Gödel who was very conscious of his own health, it had an even bigger influence.
Furtwängler was also a member of the examination committee for teaching in secondary schools. This position gave him a major amount of work since he had to hold a very large number of teacher training exams. It is remarkable that someone with his severe disability was able to carry out this high work load which would have seemed too great for a fully able person. He supervised the doctoral dissertations of 60 students including Otto Schreier and Olga Taussky-Todd. The obituary  lists 49 items by Furtwängler. These include Über das Verhalten der Ideale des Grundkörpers im Klassenkörper (1916), Punktgitter und Idealtheorie (1921), Über die linearen Mannigfaltigkeiten auf Hyperflächen zweiter Ordnung (1926), Über die simultane Approximation von Irrationalzahlen (1927), Über die simultane Approximation von Irrationalzahlen (1928), and Über affektfreie Gleichungen (1929).
Perhaps, to my [EFR] mind, his greatest achievement, however, is his proof of the principal ideal theorem which answered a conjecture made by David Hilbert. This was a major unsolved problem which many leading algebraists had failed to solve. When Furtwängler gave his proof he was 61 years old, remarkably old for someone to come up with such a major result. His 23-page paper proving the principal ideal theorem is Beweis des Hauptidealsatzes für Klassenkörper algebraischer Zahlkörper (1929). For this outstanding work he was awarded the Ernst Albe memorial prize and the medal of the Carl Zeiss Foundation in 1930. It is worth noting that this was only the second time that the Ernst Albe memorial prize had been awarded for work in pure mathematics, the previous occasion being in 1924 to Felix Klein.
Certainly these prizes were not the first honours to be given to Furtwängler. He was elected a corresponding member of the Academy of Sciences in Vienna in 1916 and became a full member in 1927. He was elected a corresponding member of the Prussian Academy of Sciences in Berlin in 1931, and a member of the German Academy of Scientists Leopoldina in Halle in 1939. In the same year of 1939, the 48th volume of Monatshefte für Mathematik und Physik was dedicated to Furtwängler on the occasion of his 70th birthday.
Furtwängler's second marriage was in 1929 to Emilie Schön. By this time he had been paralysed for over ten years but was still managing to carry out his duties despite being seriously ill. Emilie nursed him through more than ten years in a totally self-sacrificing way and helped him through his severe suffering which she made easier for him to bear. In October 1938 Furtwängler's condition forced him to retire from his duties. In January 1940 his health became even worse and everyone realised that he did not have long to live. He survived until May of that year when he suffered a stroke.
Finally, let us mention that he was writing the article Allgemeine Theorie der algebraischen Zahlen in 1938, the year in which he was forced to retire. However, his health was too poor to allow him to complete the work which was intended for the Enzyklopädie der mathematischen Wissenschaften . Furtwängler's article was completed by Martin Eichler, Helmut Hasse and Wolfram Jehne. Kenkichi Iwasawa writes in a review of the article, which finally appeared in the 1953 edition of the Encyclopaedia:-
It covers that part of algebraic number theory which is concerned with properties of finite algebraic number fields in general; class field theory, the theory of complex multiplications and other special theories on algebraic number fields are to be given in other articles. The material is well prepared and neatly arranged in a relatively small space. The main contents are as follows.
1. The ideal theory of algebraic number fields; methods of Dedekind and Kronecker.
2. Kummer-Hensel's method with valuation theory.
3. The structure of residue-class rings modulo integral ideals.
4. The ramification theory.
5. Ideal classes and units.
6. Decomposable forms and Klein's lattices.
7. Orders and their conductors.
8. Artin's conductors and Artin's L-functions.
9. Calculation of class numbers by analytic method.
Article by: J J O'Connor and E F Robertson