**Kurt Hensel**was born in East Prussia, in the city then called Königsberg. His father was Sebastian Hensel who was a landowner at the time that Kurt was born, but later moved to Berlin becoming a director of a construction firm. Kurt's mother was Julie von Adelson, the daughter of the businessman Jacob Ludwig von Adelson. It was a family containing many outstanding artists, musicians and scientists. Perhaps the most famous members of the family were Fanny and Felix Mendelssohn. Fanny was the eldest sister of the famous composer Felix Mendelssohn, and she married the Prussian court painter Wilhelm Hensel in 1829. Fanny was herself a famous pianist and composer who wrote about 500 musical compositions in all, including about 120 pieces for piano. Sebastian Hensel, the son of Fanny and Wilhelm Hensel, was born in 1830. Sebastian, who was Kurt's father, wrote a biography of the Mendelssohn family based partly on Fanny's diaries and letters, which provide a great deal of information about her brother Felix. Fanny and Felix Mendelssohn were children of Abraham Mendelssohn who was the son of the Jewish-German philosopher Moses Mendelssohn. While we are looking at this fascinating family, let us note that the youngest son of Moses Mendelssohn was Nathan Mendelssohn who was a maker of mathematical instruments. Nathan had a daughter Ottilie Ernestine who married Kummer. One of Kummer's daughters married Hermann Schwarz.

While the family lived in Königsberg, Hensel did not attend school but was taught at home by his parents up to the age of nine. After the family moved to Berlin, Hensel attended the Friedrich-Wilhelm Gymnasium and there he was taught mathematics by K H Schellbach. Schellbach was an outstanding teacher of mathematics and with such a talented pupil he was soon able to give Hensel a deep love of the subject in addition to a sound technical knowledge. There was certainly no doubt in his mind when he left the high school that the subject he wanted to study at university was mathematics. German students of this time did not choose a single university at which to study, but rather moved around to sample the courses in several different institutions. Hensel's studies were in Berlin and Bonn. Among his teachers were Lipschitz, Weierstrass, Borchardt, Kirchhoff, Helmholtz and Kronecker. It was Kronecker who was the greatest influence on Hensel and supervised his doctoral studies at the University of Berlin. Hensel submitted his thesis *Arithmetische Untersuchungen über Diskriminaten und ihre ausserwesentlichen Teiler* Ⓣ to Berlin in 1884 and he continued to work there, submitting his habilitation thesis and becoming a privatdozent in 1886.

Hensel married Gertrud Hahn in Berlin in 1887. Gertrud, who was born in 1866, was the daughter of the industrialist Albert Hahn and aunt of the educationalist Kurt Hahn. Kurt Hahn is known in Scotland as the founder (in 1934) of the internationally famous Gordonstoun School, near Elgin in the north east of the country, and known in Germany for his school at Salem. Gertrud and Kurt Hensel had one son and three daughters, including Albert, Ruth and Charlotte. Hensel was appointed to a full professorship at the University of Marburg in 1901. He spent the rest of his career there, retiring in 1930 but remaining in Marburg.

He devoted many years to the editing of Kronecker's collected works. In fact he published five volumes of Kronecker's works between the years 1895 and 1930. Note that, remarkably, these dates span Hensel's career almost exactly, with the first volume appearing in the year between his doctorate and his habilitation, and the last volume appearing in the year he retired. Two other major volumes edited by Hensel were also devoted to Kronecker's works. These are *Vorlesungen über Zahlentheorie* Ⓣ (1901), and *Vorlesungen über die Theorie der Determinanten* Ⓣ (1903). In addition Hensel wrote a eulogy for Kummer *Gedächtnissrede auf Ernst Eduard Kummer* Ⓣ and we noted above that in fact Kummer was a distant relation.

Hensel's work followed that of his doctoral supervisor Kronecker in the development of arithmetic in algebraic number fields. In 1897 the Weierstrass method of power-series development for algebraic functions led him to the invention of the *p*-adic numbers. Hensel was interested in the exact power of a prime which divides the discriminant of an algebraic number field. The *p*-adic numbers can be regarded as a completion of the rational numbers in a different way from the usual completion which leads to the real numbers. Ullrich writes in [5]:-

Hensel's invention led to the development of the concept of a field with valuation which has had a great influence on later mathematics. He was able to use his methods to prove many results in the theory of quadratic forms and number theory. Papers he published includeDuring the last decade of the19th century Kurt Hensel started his investigations on p-adic numbers ... . He was motivated by the analogies of the number field case and the function field case, e.g., by the observation that prime numbers p and linear factors z-c play similar roles in these theories. This fact had already been pointed out in articles of Kronecker(who supervised Hensel's doctorate)and of Dedekind and Heinrich Weber, which had been published in1881and1882, respectively, the paper of Kronecker based on a then unpublished manuscript from the year1858.

*Zur Theorie der algebraischen Functionen einer Veränderlichen und der Abel'schen Integrale*Ⓣ (1901),

*Über die Entwickelung der algebraischen Zahlen in Potenzreihen*Ⓣ (1901),

*Eine neue Theorie der algebraischen Zahlen*Ⓣ (1918), and

*Neue Begründung der arithmetischen Theorie der algebraischen Funktionen einer Variablen*Ⓣ (1919).

It was not until 1921 that the full potential of the *p*-adic numbers was demonstrated by Hasse when he formulated the local-global principle. He showed, at least for quadratic forms, that an equation has a rational solution if and only if it has a solution in the *p*-adic numbers for each prime *p* and a solution in the reals. It was extremely fortunate for both Hasse and for Hensel that that they came to work together. In [3] Hasse describes how he came to work under Hensel in Marburg:-

[The first book that Hensel wrote was a joint collaboration with Georg Landsberg. EntitledAfter Hecke left Göttingen]I decided to continue my studies under Kurt Hensel in Marburg .,. What prompted my decision was this. In1913Hensel published a book on number theory. I found a copy of this book at an antiquarian's in Göttingen and bought it. I found his completely new methods fascinating and worthy of thorough study. ...

*Theorie der algebraischen Funktionen einer Variablen und ihre Anwendung auf algebraische Kurven und Abelsche Integrale*Ⓣ it was published by Teubner in Leipzig in 1902. It was reprinted by the Chelsea Publishing Company of New York in 1965. Another book of major importance was Hensel's

*Theorie der algebraischen Zahlen*Ⓣ published in 1908. It was in this book that he developed his great idea of

*p*-adic numbers into a systematic theory. He continued to develop his theory in another book

*Zahlentheorie*Ⓣ published in 1913, the book which was mentioned in Hasse's quote above and the one which led to him becoming Hensel's student. In these two books he showed the power of applying

*p*-adic methods to he theory of divisibility in algebraic number fields. Not only is the term

*p*-adic integer due to Hensel but also in

*Zahlentheorie*he uses the description "Fermat's Little Theorem" for the first time:-

From 1901 Hensel was editor of the prestigious and influential Crelle's Journal. He was honoured in 1931 with the award of an honorary doctorate from the university of Oslo.There is a fundamental theorem holding in every finite group, usually called Fermat's Little Theorem because Fermat was the first to have proved a very special part of it.

Hensel died in the summer of 1941, in the middle of World War II. A year later his daughter-in-law sold over one hundred items from his mathematical library to the Reichs-Universitat Strassburg.

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

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