**Alfred Loewy**'s father was Hermann Loewy and his mother was Lina Löwental. It was a Jewish family and Alfred was brought up in a strict orthodox manner. Alfred was born two years after the unification of Germany, and with that came full emancipation of Jews. It might have looked as if this was a good beginning, but sadly for the Loewy family anti-Semitism became an organized movement in Germany with its own political parties. Discrimination against Jewish families, especially those with an orthodox lifestyle, became widespread and Loewy would encountered increasing discrimination throughout his life.

Loewy studied at the universities of Breslau, Munich, Berlin and Göttingen between 1891 and 1895. He was awarded a doctorate by the University of Munich in 1894 for his thesis *Über die Transformation einer quadratischen Form in sich selbst mit Anwendungen auf die Linien- und Kugelgeometrie* Ⓣ which he had written with C L Ferdinand Lindemann and Gustav Bauer as his advisors. Then, from 1897, Loewy taught at the University of Freiburg having submitted his habilitation thesis. His work at that time concerned the theory of linear substitutions and he published two papers *Zur Theorie der linearen Substitutionen* Ⓣ in 1896 and 1897. He was appointed as an extraordinary professor at Freiburg in 1902. This made him secure enough financially to marry and in that year he married Therese Neuburger. Loewy became an honorary ordinary professor at Freiburg in 1916 before his appointment as ordinary professor in 1919. He was thesis advisor to a number of famous students, in particular Wolfgang Krull, who was awarded his doctorate in 1922, and Friedrich Karl Schmidt, who was awarded his doctorate in 1925. Other algebraists who spent some time in Freiburg working under Loewy are E Witt, Bernhard Neumann, R Brauer, R Baer, and A Scholz.

Anti-Semitism increased in Germany following the end of World War I. Anti-Semites joined forces with nationalists in attempting to blame the Jews for Germany's defeat. Increasing discrimination was not the only source of difficulty in Loewy's life. Already by 1916 he had lost the sight of one eye. His eyesight began to fail completely from about 1920 and he became totally blind before his death after a failed operation in 1928 left his other eye completely blind also. Despite these severe health problems Loewy continued to carry out his teaching duties. He could battle against blindness and against the hurt of anti-Semitism directed at him, but the final blow came in 1933 when anti-Semitism became part of the law of the land. On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law was passed that provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Loewy was forced to retire in 1933 under the Civil Service Law.

Loewy worked on linear groups, the algebraic theory of differential equations and actuarial mathematics. He published 70 papers and a number of books. Among his papers published in *Mathematische Annalen* are: *Über bilineare Formen mit conjugirt imaginären Variablen* Ⓣ (1898); *Über die Charakteristik einer reelen quadratischen Form von nicht verschwindender Determinante* Ⓣ (1899); *Zur Theorie der Gruppen linearer Substitutionen* Ⓣ (1900); *Über eine besondere Gattung endlicher discreter Gruppen* (1901); *Zur Theorie der endlichen continuirlichen Transformationsgruppen* Ⓣ (1901); and *Über reduzible lineare homogene Differentialgleichungen* Ⓣ (1903). He also published papers (in German) in the *Transactions of the American Mathematical Society* such as: *On the reducibility of real groups of linear homogeneous substitutions* (1903); *On group theory, with applications to the theory of linear homogeneous differential equations* (1904); and *On completely reducible groups that belong to a group of linear homogeneous substitutions* (1905). He also edited German translations of works by Abel, Fourier, Charles Sturm and Étienne Pascal.

Among Loewy's most famous books are *Lehrbuch der Algebra* Ⓣ (1915) and *Mathematik des Geld- und Zahlungsverkehrs* Ⓣ (1920). The first of these was one of the first works to introduce into Germany the methodology, the terminology and the achievements of postulational analysis as it was being developed in the United States.

Let us now mention the connection between Loewy and Fraenkel. Loewy was Fraenkel's uncle by marriage and he exerted a large influence on Fraenkel's career in its early stages. It was Loewy who persuaded Fraenkel to travel to Marburg to study under Hensel and it was Loewy who had helped Fraenkel publish his early work in Crelle's journal with a paper about the date of Easter. But the mathematical topics Fraenkel studied were also influenced by Loewy whose interest in the study of axiomatic systems encouraged a similar interest by Fraenkel. The relationship worked both ways round, however, and Loewy's *Grundlagen der Arithmetik* Ⓣ, published in 1915, was prepared with Fraenkel's assistance. Loewy mentioned in this work that in the system of integers, the product of any two integers is zero, if and only if one of them is zero. Such ideas clearly influenced Fraenkel to introduce the notion of a ring, and in particular zero-divisors in rings.

**Article by:** *J J O'Connor* and *E F Robertson*

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