Professor Albrecht Fröhlich

Albrecht Fröhlich, mathematician: born Munich 22 May 1916; Assistant Lecturer, University College, Leicester 1950-1952; Lecturer, University College of North Staffordshire 1952-55; Reader in Pure Mathematics, King's College London 1955-62, Professor 1962-81 (Emeritus), Head of Department of Mathematics, 1969-81; Senior Research Fellow, Imperial College London, 1982-96; Fellow, Robinson College, Cambridge 1982-2001; FRS 1976; married 1950 Ruth Brooks (one son, one daughter); died Cambridge 8 November 2001.

"The mathematic's patterns, like the painter's or the poet's, must be beautiful . . . beauty is the first test"; so wrote G.H. Hardy in his A Mathematician's Apology (1940). Albrecht Fröhlich was an outstanding mathematician, whose influence on the development of algebraic number theory has been profound and for whom the beauty of its ideas and methods was always the first test.

Ali Fröhlich was the youngest of the three children of Julius and Frida Fröhlich, from Rexingen in the Black Forest, a village where a high proportion of the residents, like the Fröhlichs, were Jewish. Ali was born in Munich and he grew up in the more hopeful period of the Weimar Republic, but, when he was 17, his father was beaten up by a gang of Nazi Brownshirts, who then came looking for Ali; his mother called the police, who were not yet dominated by the Nazis, and who wisely arrested him as an "enemy of the Third Reich" – only to release him the following day. His sister, Betti, was already living in Palestine; after a year in Alsace, Fröhlich and his parents joined her there, where he worked for 12 years, as road builder, plumber and electrician in a railway workshop.

Fröhlich's elder brother, Herbert, was a physicist who, in 1945, was beginning an academic career at Bristol University. Herbert proposed that his brother should come to Bristol to read Mathematics, even though he had no academic qualifications. That he was allowed to do so showed great vision on the part of the university and Ali Fröhlich began his first year as an undergraduate, albeit a term late. He graduated in 1948 with first class Honours. It was whilst he was an undergraduate at Bristol that he met Ruth Brooks, a medical student, whom he married in 1950.

He completed his PhD in 1951, on problems concerning Galois extensions of number fields, work whose subsequent development by others was a source of joy to him. He served as lecturer at the University Colleges of Leicester and North Staffordshire, and then, in 1955, was appointed to a Readership in Mathematics at King's College London, and to a Chair in 1962; he was also Head of Department from 1969 until his retirement in 1981.

In his apologia, Hardy also wrote, "I do not know of an instance of a major mathematical advance initiated by a man past 50." Yet Fröhlich was over 50 when he produced work of astonishing originality and significance. He made important contributions in abstract algebra and its applications to the higher arithmetic.

One very significant contribution, which served to establish a strong school of algebraic number theory in Britain and to develop the subject internationally, was to the organisation and presentation of the "Brighton" conference on Algebraic Number Theory in 1965. The conference opened the eyes of many to the meaning and beauty of "class field theory" and the Proceedings remain a standard reference. A central problem in the application of algebra to arithmetic is to generalise familiar ideas, such as prime numbers, to extensions of the number system. The laws by which such extensions are built up are mysteriously hidden in the structure of the numbers from which one starts and class field theory unfolds the mystery.

In the 1970s Fröhlich proved a completely unexpected relationship between two apparently unrelated mathematical structures. The first concerns the way in which extensions of the rational numbers and the integers to more general "algebraic number fields" can be expressed in terms of the integers (the so-called integral bases); the second concerns a "root number", associated with a product resembling prime factorisation, called an "L-function". Professor Jean-Pierre Serre of the Collège de France had suggested the "crazy idea" – "trop beau pour être vrai" – that the first construction, of a "normal integral basis", is possible or not according as the root-number is plus 1 or minus 1. Fröhlich's proof of the "crazy idea" was beautiful and its inner coherence so natural that it received Serre's ready acclamation.

Honours and invitations to Visiting Professorships followed. Fröhlich was elected a Fellow of the Royal Society in 1976, and was especially proud of the fact that he and his brother were one of the few sibling pairs of Fellows since the Royal Society received its Charter in 1662. He was awarded the Senior Berwick Prize of the London Mathematical Society.

He continued to receive further honours, including honorary doctorates from the University of Bordeaux and his own Bristol University, and in 1992 was awarded the premier British mathematical prize: the De Morgan Medal of the London Mathematical Society. His work continued to inspire other mathematicians; he sometimes referred to that time as a "period of grace".

In 1982 he was elected to Fellowship at Robinson College, Cambridge, where he and his wife enjoyed the warmth of its corporate life and his 70th and 80th birthday celebrations there will long be remembered. His retirement freed him from administration and he was able to travel, and to engage in original work up to the age of 82, when declining health and impaired vision brought to an end his Indian Summer. But even then he tried to keep abreast of mathematical developments, and attended a seminar in Cambridge two weeks before his death.

Ali Fröhlich was an exceptionally charming and warm man; his integrity and kindness and his sense of humour, together with his infectious enthusiasm for mathematics, qualified him admirably for leadership in research and for being an understanding and supportive head of department.

He retained a great fondness for Germany and loved to attend mathematical conferences at Oberwolfach in the Black Forest. To walk and talk with him on a Spaziergang there, or indeed on Wimbledon Common, for he was always and everywhere the same, was an unforgettable privilege. He loved music and played the piano and his sense of humour and his humility were apparent in that too: he used to say that he disliked the music of Tchaikovsky and loved that of Brahms – but couldn't always tell the difference.

Vernon Armitage