Hofreiter went to school in Linz and studied from 1923 in Vienna with Hans Hahn, Wilhelm Wirtinger, Emil Müller at the Technische Universität Wien on descriptive geometry, and Philipp Furtwängler, with whom he obtained his doctorate in 1927 on the reduction theory of quadratic forms (Eine neue Reduktionstheorie für definite quadratische Formen).
His dissertation and habilitation thesis dealt with the reduction theory of quadratic forms, which Gauss, Charles Hermite and Hermann Minkowski had worked on previously.
In 1934, he proved the existence of infinitely many real quadratic number fields without a Euclidean algorithm.
In addition to their work at the Aviation Research Institute, Gröbner and Hofreiter continued to give lectures and seminars at the Technical University of Braunschweig.
His doctoral students included Edmund Hlawka, Peter Gruber and Leopold Schmetterer.