库尔特·亨泽尔Kurt Hensel

库尔特·亨泽尔 出生于东普鲁士，当时称为柯尼斯堡的城市。他的父亲是 Sebastian 亨泽尔，在 亨泽尔 出生时是一位地主，但后来搬到柏林，成为一家建筑公司的董事。亨泽尔 的母亲是 Julie von Adelson，是商人 Jacob Ludwig von Adelson 的女儿。这是一个包含许多杰出艺术家、音乐家和科学家的家庭。也许这个家族最著名的成员是 Fanny 和 Felix Mendelssohn。Fanny 是著名作曲家 Felix Mendelssohn 的长姐，她于 1829 年嫁给了普鲁士宫廷画家 Wilhelm Hensel。Fanny 本人是一位著名的钢琴家和作曲家，总共创作了约 500 部音乐作品，其中包括约 120 首钢琴曲。Sebastian 亨泽尔，Fanny 和 Wilhelm Hensel 的儿子，出生于 1830 年。Sebastian，即 亨泽尔 的父亲，部分基于 Fanny 的日记和信件，写了一部 Mendelssohn 家族的传记，这些日记和信件提供了大量关于她弟弟 Felix 的信息。Fanny 和 Felix Mendelssohn 是 马克斯·亚伯拉罕 Mendelssohn 的孩子，而后者是犹太-德国哲学家 Moses Mendelssohn 的儿子。在我们审视这个迷人的家族时，让我们注意到 Moses Mendelssohn 的最小的儿子是 Nathan Mendelssohn，他是一位数学仪器制造者。Nathan 有一个女儿 Ottilie Ernestine，她嫁给了 恩斯特·爱德华·库默尔。恩斯特·爱德华·库默尔 的一个女儿嫁给了 赫尔曼·阿曼杜斯·施瓦茨。

当家庭住在柯尼斯堡时，亨泽尔没有上学，而是由父母在家教到九岁。家庭搬到柏林后，亨泽尔上了Friedrich-Wilhelm 文理中学（Gymnasium），在那里由K H Schellbach教数学。Schellbach是一位杰出的数学教师，有了这样一个有天赋的学生，他很快就能够让亨泽尔对这个科目产生深厚的热爱，同时获得扎实的技术知识。当他离开高中时，他心中毫无疑问，他想在大学学习的科目是数学。这个时代的德国学生不选择单一大学学习，而是四处走动，在不同的机构中体验课程。亨泽尔的学习在柏林和波恩进行。他的老师包括鲁道夫·利普希茨、卡尔·魏尔斯特拉斯、卡尔·威廉·博尔夏特、古斯塔夫·基尔霍夫、赫尔曼·冯·亥姆霍兹和利奥波德·克罗内克。是利奥波德·克罗内克对亨泽尔影响最大，并在柏林大学指导他的博士研究。亨泽尔于1884年向柏林提交了他的学位论文Arithmetische Untersuchungen über Diskriminaten und ihre ausserwesentlichen Teiler Ⓣ（关于判别式及其非本质因子的算术研究），他继续在那里工作，提交了他的教授资格论文（Habilitation）论文，并于1886年成为privatdozent。

亨泽尔于1887年在柏林与Gertrud Hahn结婚。Gertrud生于1866年，是工业家亚伯拉罕·阿德里安·艾伯特 Hahn的女儿，也是教育家亨泽尔 Hahn的姑姑。亨泽尔 Hahn在苏格兰以1934年创办国际著名的Gordonstoun学校而闻名，该校位于该国东北部的Elgin附近，在德国则以他在拉斐尔·塞勒姆的学校而闻名。Gertrud和亨泽尔有一个儿子和三个女儿，包括艾伯特、Ruth和Charlotte。亨泽尔于1901年被任命为马尔堡大学的正教授。他在那里度过了余下的职业生涯，1930年退休但仍留在马尔堡。

他投入多年编辑利奥波德·克罗内克的文集。事实上，他在1895年至1930年间出版了五卷利奥波德·克罗内克的著作。值得注意的是，这些日期几乎正好跨越了亨泽尔的职业生涯，第一卷出现在他博士和博士后资格之间的那一年，最后一卷出现在他退休的那一年。亨泽尔编辑的另外两卷主要著作也致力于利奥波德·克罗内克的著作。这些是Vorlesungen über Zahlentheorie Ⓣ（数论讲义）（1901年）和Vorlesungen über die Theorie der Determinanten Ⓣ（行列式理论讲义）（1903年）。此外，亨泽尔为恩斯特·爱德华·库默尔写了一篇悼词Gedächtnissrede auf Ernst Eduard Kummer Ⓣ（关于恩斯特·爱德华·库默尔的纪念演讲），我们上面注意到，事实上恩斯特·爱德华·库默尔是一个远亲。

亨泽尔的工作追随其博士导师利奥波德·克罗内克在代数数域中算术的发展。1897年，卡尔·魏尔斯特拉斯的代数函数幂级数展开方法促使他发明了$p$进数。亨泽尔对整除一个代数数域的判别式的素数的精确幂感兴趣。$p$进数可以被视为有理数的一种完备化，与导致实数通常的完备化方式不同。乌尔里希在[5]中写道：-

在19世纪的最后十年，亨泽尔开始了对p进数的研究……。他受到数域情形和函数域情形类比的启发，例如，观察到素数p和线性因子z-c在这些理论中扮演类似的角色。这一事实已经在利奥波德·克罗内克（他指导了亨泽尔的博士学位）以及理查德·戴德金和海因里希·马丁·韦伯的文章中指出，这些文章分别于1881年和1882年发表，利奥波德·克罗内克的论文基于当时未发表的手稿，来自1858年。

亨泽尔的发明导致了赋值域概念的发展，这对后来的数学产生了巨大影响。他能够使用他的方法证明二次型理论和数论中的许多结果。他发表的论文包括Zur Theorie der algebraischen Functionen einer Veränderlichen und der Abel'schen Integrale Ⓣ（关于单变量代数函数和Abel积分的理论）（1901年），Über die Entwickelung der algebraischen Zahlen in Potenzreihen Ⓣ（关于代数数展开为幂级数）（1901年），Eine neue Theorie der algebraischen Zahlen Ⓣ（代数数的新理论）（1918年），以及Neue Begründung der arithmetischen Theorie der algebraischen Funktionen einer Variablen Ⓣ（单变量代数函数算术理论的新基础）（1919年）。

直到1921年，$p$进数的全部潜力才由哈塞展示出来，当时他提出了局部-整体原则。他表明，至少对于二次型，一个方程有有理数解当且仅当它对每个素数$p$都有$p$进数解，并且有实数解。对于哈塞和亨泽尔来说，他们能够一起工作是非常幸运的。在[3]中，哈塞描述了他如何在马尔堡的亨泽尔指导下工作：-

[在埃里希·赫克离开哥廷根之后]我决定在马尔堡继续跟随亨泽尔学习。促使我做出这一决定的原因是这样的。1913年，亨泽尔出版了一本关于数论的书。我在哥廷根的一家旧书店里发现了这本书，便买了下来。我发现他那些全新的方法引人入胜，值得深入研究。……

亨泽尔写的第一本书是与格奥尔格·兰茨贝格合作完成的。书名为Theorie der algebraischen Funktionen einer Variablen und ihre Anwendung auf algebraische Kurven und Abelsche IntegraleⓉ（《单变量代数函数论及其在代数曲线和阿贝尔积分上的应用》），1902年由莱比锡的Teubner出版社出版。1965年由纽约的Chelsea出版公司重印。另一本极为重要的书是亨泽尔的Theorie der algebraischen ZahlenⓉ（《代数数论》），1908年出版。正是在这本书中，他把自己关于$p$进数的伟大思想发展成了一套系统的理论。他在另一本书ZahlentheorieⓉ（《数论》）中继续发展他的理论，该书于1913年出版，就是上文哈塞引语中提到的那本书，也是促使他成为亨泽尔的学生的那本书。在这两本书中，他展示了将$p$进方法应用于代数数域中的整除性理论的威力。不仅$p$进整数这一术语出自亨泽尔，而且在Zahlentheorie中他首次使用了“皮埃尔·德·费马小定理”这一说法：——

有一个在每个有限群中都成立的基本定理，通常被称为费马小定理，因为皮埃尔·德·费马是第一个证明了其中某个非常特殊的部分的人。

从1901年起，亨泽尔担任享有盛誉且极具影响力的奥古斯都·利奥波德·克雷勒杂志的编辑。1931年，他荣获奥斯陆大学授予的名誉博士学位。

亨泽尔于1941年夏天去世，正值第二次世界大战期间。一年后，他的儿媳将他数学图书馆中的一百多件物品卖给了斯特拉斯堡帝国大学。

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]:-

During the last decade of the 19th 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 in 1881 and 1882, respectively, the paper of Kronecker based on a then unpublished manuscript from the year 1858.

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 include 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:-

[After Hecke left Göttingen] I decided to continue my studies under Kurt Hensel in Marburg .,. What prompted my decision was this. In 1913 Hensel 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. ...

The first book that Hensel wrote was a joint collaboration with Georg Landsberg. Entitled 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:-

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.

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.

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.