He wrote an 1896 doctoral dissertation at the University of Göttingen on cubic forms (Zur Theorie der in Linearfaktoren zerlegbaren ganzzahlingen ternären kubischen Formen), under Felix Klein. Most of his academic life, from 1912 to 1938, was spent at the University of Vienna, where he taught for example Kurt Gödel, who later said that Furtwängler's lectures on number theory were the best mathematical lectures that he ever heard; Gödel had originally intended to become a physicist but turned to mathematics partly as a result of Furtwängler's lectures. Furtwängler was paralysed and, without notes, lectured from a wheelchair while his assistant wrote equations on the blackboard. Some of Furtwängler's doctoral students were Wolfgang Gröbner, Henry Mann, Otto Schreier, and Olga Taussky-Todd.
He is now best known for his contribution to the principal ideal theorem in the form of his Beweis des Hauptidealsatzes für Klassenkörper algebraischer Zahlkörper (1929).