伦纳德·尤金·迪克森Leonard Eugene Dickson

伦纳德·尤金·迪克森，或常被称为L E 迪克森，出生于艾奥瓦州，但由于他年幼时全家搬到了得克萨斯州，他一直认为自己是个得克萨斯人。他的父母是Lucy Tracy和Campbell 迪克森，后者是一名银行家。Campbell还是一名商人，通过投资房地产赚钱。迪克森在家乡Cleburne上了小学和中学。他进入得克萨斯大学，很快受到G·B·豪斯泰德的影响，后者鼓励他学习数学。迪克森在数学领域广泛学习，但专攻G·B·豪斯泰德自己的欧几里得和non-euclidean geometry科目。迪克森于1893年获得学士学位，1894年获得硕士学位，同样是在G·B·豪斯泰德的指导下。

迪克森申请了哈佛和芝加哥的博士奖学金。他接受了哈佛的录取通知，但在后来收到芝加哥的录取通知后改变了主意。在芝加哥，他由伊莱基姆·黑斯廷斯·穆尔指导，但那里的其他人也影响了他，例如Bolza和海因里希·马什克。迪克森于1896年获得芝加哥大学博士学位，学位论文题为The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group。这是芝加哥授予的第一个数学博士学位。

迪克森随后在莱比锡与索菲斯·李共事一段时间，后来在巴黎与卡米耶·若尔当共事。回到美国后，他在乔治·伯克利的加利福尼亚大学担任讲师。他于1899年被任命为得克萨斯大学奥斯汀分校的副教授。然而，伊莱基姆·黑斯廷斯·穆尔和他在芝加哥的同事们希望迪克森回到那里，他们为他提供了一个永久教职。他立即接受了，并于1900年至1907年在芝加哥大学担任助理教授，然后担任副教授至1910年，当时他被提升为正教授。他在芝加哥大学担任教授直至职业生涯结束，1939年退休时被授予荣休教授。在这些年里，他确实有段时间离开芝加哥，主要是在加利福尼亚大学，他在1914年、1918年和1922年担任访问教授。迪克森于1902年与Susan McLeod Davis结婚；他们有两个孩子。

迪克森的数学成果极为丰硕，其已发表著作目录包含275项。他研究有限域，并扩展了由约瑟夫·韦德伯恩和埃利·嘉当开创的线性结合代数理论。他在数论中证明了许多有趣的结果，利用伊万·维诺格拉多夫的结果，在研究加性数论时推导出理想爱德华·华林定理。

1901年，他的著名著作Linear groups with an exposition of the Galois field theory出版了。也许首先要说明的是，这本书由莱比锡的Teubner出版，部分原因可能是菲利克斯·克莱因的建议，但主要是因为当时没有成熟的美国科学出版商。这本书是他1896年博士论文的修订和扩充版。然而，我们应该注意到，在出版这本书之前，迪克森在之前的五年里已经发表了43篇研究论文，其中除了七篇之外，全部是关于有限线性群的。在给菲利克斯·克莱因的书籍提案中，迪克森写道：-

这里宣布的这本书提议处理线性同余群，或者更一般地，处理Galois field中的线性群，这一主题因埃瓦里斯特·伽罗瓦、恩里科·贝蒂、Mathieu [以米里迂·拉·马丢]、卡米耶·若尔当以及许多近期作者的努力而丰富。

在给菲利克斯·克莱因的信中，迪克森还谈到：-

...引入显著的简化...

和

...呈现理论的某些部分，而不给出已发表论文中的困难计算。

Parshall 在20中描述这本书时写道：-

迪克森提出了经典线性群的一个统一、完整且一般的理论——不仅仅像卡米耶·若尔当所做的那样在素域GF(p)上——而是在一般有限域GF(pn)上，并且他是在这些基础域已有成熟理论的背景下做到这一点的。……他的书代表了数学文献中对有限域的首次系统处理。

除Linear groups with an exposition of the Galois field theory外，迪克森还出版了17本书。三卷本的History of the Theory of Numbers（1919—23）是另一部至今仍被广泛查阅的著名著作。这三卷涵盖：整除性与素性；丢番图分析；以及二次型与高次型。这部著作几乎没有解释，也没有试图为所描述的结果构建背景，但它本质上包含了从数学起源直到20世纪20年代的每一个数论思想。德里克·亨利·莱默（也许相当苛刻地）将其描述为：-

……一份可以据以撰写历史的参考文献列表。

亚伯拉罕·阿德里安·艾伯特在[3]中指出，迪克森的三卷本著作：-

……对一个更普通的人来说，仅凭它本身就足以成为毕生的工作。

尽管其内容全面，二次互反律并未被讨论，其原因在[14]中得到了解释，其中给出了计划中的第四卷的细节，该卷从未出版。在[12]中，考虑了为什么迪克森在其研究能力的巅峰时期选择将如此多的时间投入到这个项目中的问题。给出了三个原因：他只是想知道在这个主题上已经完成的所有工作；他想创作一部美国著作，详细报告一个数学主题，与欧洲正在编写的类似报告相匹配；最后是迪克森自己给出的原因，即它：-

……符合我的信念，即每个人都应该在一生中的某个时候致力于做一些严肃有用的工作，而除了从中获得的满足感之外，极不可能有任何其他回报。

迪克森于1939年出版了Modern Elementary Theory of Numbers。Brinkmann在评论这本书时写道：-

本书的前四章简要但令人满意地介绍了数论的通常初等主题，包括二元二次型的简短叙述。这部分内容初学者可以轻松阅读，并且有许多适合此类读者的问题。剩下的两百页涉及更高级的主题，其中大多数具有“加性”特征。证明是“初等的”（附录除外），并且通常比文献中的更简单。有几章涉及二次型。其中一章包含关于将自然数表示为三个平方和的经典定理的推广。……

我们应该审视的迪克森研究的最后一个领域是他在代数方面的工作。伊莱基姆·黑斯廷斯·穆尔和迪克森在域方面做了大量工作，但正是1904年约瑟夫·韦德伯恩来到芝加哥时，那里开始了有限可除代数的工作。1905年，约瑟夫·韦德伯恩证明了有限可除代数的交换性，但[21]对迪克森在这项工作中的互动提供了很多见解。事实上，约瑟夫·韦德伯恩发现了一个证明，后来被认为包含一个漏洞，但当时没有人注意到这一点。迪克森起初寻找反例，却自己找到了一个（相当晦涩的）证明，并向约瑟夫·韦德伯恩展示了它。然后约瑟夫·韦德伯恩利用迪克森证明中的思想设计了另外两个优雅的证明。迪克森对反例的寻找使他考虑非结合代数，并在一系列论文中确定了域上所有三维和四维（非结合）可除代数。他出版了主要著作Linear algebras（1914年）、Algebras and their arithmetics（1923年）和Modern algebraic theories（1926年）。

亚伯拉罕·阿德里安·艾伯特在[9]中写道：-

迪克森是一位鼓舞人心的教师。

然而Parshall写道[22]：-

尽管在课堂上并非特别有天赋，迪克森作为导师巧妙地激发了芝加哥大学研究生对他工作的兴趣。

事实上，Fenster在[17]中提出了一个问题：当迪克森被公认为不特别擅长课堂教学时，他如何能吸引67名博士生进入代数和数论研究[17]：-

他的讲座简洁而未加修饰，对学生说话严厉。他经常指定教科书（往往是他自己的）中的阅读材料，要么叫学生展示和分析材料，要么整节课都由他讲授。……然而，鉴于迪克森对学生数学弱点的不容忍，他的评论可能很严厉，即使并非针对个人。他并不旨在让学生自我感觉良好。

她对他为何如此成功的回答是，他作为领先研究数学家的榜样，这是他的学生希望效仿的，而不是因为他作为导师的角色。也许他在接收学生之前要求如此严格，这一事实对他有利而非不利[17]：-

迪克森 对他的未来博士生有一个突然的死亡测试：他分配一个比学位论文问题更短的预备问题，如果学生能在三个月内解决它，迪克森 就会同意指导该研究生的工作。否则，学生必须另找导师。

迪克森 获得了许多荣誉。他于1913年当选为 国家科学院（美国），也是美国哲学学会、American Academy of Arts and Sciences、London Mathematical Society、French Academy of Sciences 和 捷克数学家和物理学家联合会 的成员。美国科学促进会决定设立一个奖项，以表彰对科学进步做出的最重大贡献。迪克森 是该奖项的第一位获得者，因其在代数算术方面的工作于1924年获得1000美元奖金。他还是 美国数学会 于1928年颁发的 法兰克·尼尔森·寇尔 代数奖的第一位获得者，获奖作品是他1927年在苏黎世和莱比锡出版的 Algebren und ihre Zahlentheorie。迪克森 与 美国数学会 关系密切，于1917-1918年担任其主席，此前于1913年担任其学术讨论会讲师。他于1918年12月发表了主席演讲，主题是 Mathematics in War Perspective，在演讲中他批评美国在数学准备方面未能达到英国、法国和德国的水平。他说：-

但愿不再可能出现这样的情况：成千上万的年轻人由于缺乏足够的数学准备，在陆军和海军工作中受到如此严重的妨碍。

作为对授予 迪克森 荣誉的最后评论，我们注意到普林斯顿（1941年）和哈佛（1936年）是授予他荣誉学位的大学之一。

我们在本文开头提到，迪克森一直认为自己是个德克萨斯人，因此他退休后回到“家”是很自然的。他整个职业生涯的研究活动水平高于大多数数学家所能想象，也许正因为如此，他在退休的十五年中实际上放弃了研究。

关于 迪克森 的性格，[22] 中描述如下：-

迪克森 是一个坚韧不拔的人，倾向于直率地说出自己的想法；他对他人工作的赞扬总是很吝啬。……他沉迷于桥牌和台球的强烈爱好，据说他不喜欢在这两种游戏中输掉。

他无疑对美国的数学产生了重大影响，很大程度上正是因为他，美国在代数研究中的角色发生了转变。

Leonard Dickson, or L E Dickson as he was often called, was born in Iowa but since his family moved to Texas when he was a young child he always considered himself a Texan. His parents were Lucy Tracy and Campbell Dickson, who worked as a banker. Campbell was also a merchant and made money through investing in real estate. Leonard attended both primary and secondary school in his home town of Cleburne. He entered the University of Texas and quickly came under the influence of Halsted who encouraged him to study mathematics. Dickson studied widely within mathematics but specialised in Halsted's own subjects of euclidean and non-euclidean geometry. Dickson received his B.S. in 1893 and his M.S. in 1894, again under Halsted's supervision.

Dickson applied for doctoral fellowships at both Harvard and Chicago. He accepted an offer from Harvard but, on receiving a later offer from Chicago, changed his mind. At Chicago he was supervised by Eliakim Moore, but others there influenced him, for example Bolza and Maschke. Dickson received a Ph.D. from the University of Chicago in 1896 for a dissertation entitled The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group. It was the first mathematics doctorate awarded by Chicago.

Dickson then spent some time with Lie at Leipzig and later with Jordan in Paris. On returning to the United States he became an instructor at the University of California in Berkeley. He was appointed as associate professor at the University of Texas at Austin in 1899. However Eliakim Moore and his colleagues in Chicago were keen that Dickson should return there and they offered him a permanent post on the faculty. He accepted immediately and served as assistant professor at the University of Chicago from 1900 to 1907, then associate professor to 1910 when he was promoted to full professor. He remained as a professor at Chicago for the rest of his career, retiring in 1939 when he was made professor emeritus. He did spend periods during these years away from Chicago, principally at the University of California where he was a visiting professor in 1914, 1918, and 1922. Dickson married Susan McLeod Davis in 1902; they had two children.

Dickson's mathematical output was vast and his list of published works contains 275 items. He worked on finite fields and extended the theory of linear associative algebras initiated by Wedderburn and Cartan. He proved many interesting results in number theory, using results of Vinogradov to deduce the ideal Waring theorem in his investigations of additive number theory.

In 1901 his famous book Linear groups with an exposition of the Galois field theory was published. Perhaps the first comment to make is that it was published by Teubner of Leipzig, probably partly because of Klein's advice, but mainly because there was no well-established American scientific publisher. The book was a revised and expanded version of his 1896 doctoral thesis. However we should note that before publishing the book, Dickson had already published 43 research papers in the preceding five years which, with the exception of seven, were all on finite linear groups. In the proposal for his book, sent to Klein, Dickson wrote:-

The book here announced proposes to treat of linear congruence groups, or more generally, of linear groups in a Galois field, a subject enriched by the labors of Galois, Betti, Mathieu [Émile Mathieu], Jordan and many recent writers.

In his letter to Klein, Dickson also talks of:-

... introducing marked simplifications ...

and

... presenting parts of the theory without the difficult calculations given in the published papers.

Parshall in [20] describing the book writes:-

Dickson presented a unified, complete, and general theory of the classical linear groups - not merely over the prime field GF(p) as Jordan had done - but over the general finite field GF(pn ), and he did this against the backdrop of a well-developed theory of these underlying fields. ... his book represented the first systematic treatment of finite fields in the mathematical literature.

Dickson published 17 books in addition to Linear groups with an exposition of the Galois field theory. The 3-volume History of the Theory of Numbers (1919-23) is another famous work still much consulted today. The three volumes cover: Divisibility and primality; Diophantine analysis; and Quadratic and higher forms. The work contains little interpretation and makes no attempt to build a context for the results being described, yet it contains essentially every number theoretic idea from the very beginning of mathematics up to the 1920s. Derrick Lehmer described it (perhaps rather harshly) as:-

... list of references from which a history might be written.

Abraham Albert remarks in [3] that Dickson's three volume work:-

... would be a life's work by itself for a more ordinary man.

Despite its comprehensive nature, quadratic reciprocity is not discussed and the reason for this is explained in [14] where details of a planned fourth volume, which was never published, are given. In [12] the question of why Dickson chose to devote so much time to this project when at the height of his research powers is considered. Three reasons are given: he simply wanted to know all of the work which had been done in the subject; he wanted to create an American work reporting in detail on a mathematical subject which matched similar reports being compiled in Europe; and finally the reason which Dickson gave himself, namely that it:-

... fitted in with my conviction that every person should aim to perform at some time in his life some serious useful work for which it is highly improbable that there will be any reward whatever other than his satisfaction therefrom.

Dickson published Modern Elementary Theory of Numbers in 1939. Brinkmann, reviewing this book, writes:-

The first four chapters of this book furnish a brief but satisfactory introduction to the usual elementary topics of number theory, including a short account of binary quadratic forms. This part of the book can be easily read by a beginner and there are many problems suitable for such a reader. The remaining two hundred pages deal with more advanced subjects, most of them "additive" in character. The proofs are "elementary" (except for the Appendix) and are, in general, simpler than those in the literature. There are several chapters dealing with quadratic forms. One of these contains generalizations of the classical theorem on representing a natural number as the sum of three squares. ...

The final area of Dickson's research which we should look at is his work on algebras. Eliakim Moore and Dickson had done much work on fields but it was the arrival of Wedderburn in Chicago in 1904 when work began there on finite division algebras. In 1905 Wedderburn proved the commutativity of finite division algebras but [21] gives much insight into the interactions of Dickson in this work. In fact Wedderburn discovered a proof which was subsequently seen to contain a gap but nobody noticed this at the time. Dickson, who at first looked for a counterexample, was led to find a (rather obscure) proof himself which he showed to Wedderburn. Then Wedderburn devised two further elegant proofs using ideas from Dickson's proof. Dickson's search for a counterexample led him to consider non-associative algebras and in a series of papers he determined all three and four-dimensional (non-associative) division algebras over a field. He published the major texts Linear algebras in 1914, Algebras and their arithmetics in 1923, and Modern algebraic theories in 1926.

Abraham Albert writes in [9] that:-

Dickson was an inspiring teacher.

However Parshall writes [22]:-

Although not especially gifted in the classroom, Dickson adeptly engaged the interests of the graduate students at the University of Chicago in his work as an advisor.

In fact Fenster in [17] asks the question of how Dickson could attract 67 doctoral students into research in algebra and number theory when he was recognised as not particularly good at classroom teaching [17]:-

He delivered terse and unpolished lectures and spoke sternly to his students. He frequently assigned readings from a textbook (often one of his own) and he either called on students to present and analyse the material or he lectured the entire hour. ... Given Dickson's intolerance for student weaknesses in mathematics, however, his comments could be harsh, even though not intended to be personal. He did not aim to make students feel good about themselves.

Her answer to why he was so successful is by his example as a leading research mathematician, something his students wished to emulate, rather than because of his role as an advisor. Perhaps the fact that he was so demanding before accepting students worked in his favour rather than against him [17]:-

Dickson had a sudden death trial for his perspective doctoral students: he assigned a preliminary problem which was shorter than a dissertation problem, and if the student could solve it in three months, Dickson would agree to oversee the graduate student's work. If not the student had to look elsewhere for an advisor.

Dickson was awarded many honours. He was elected to the National Academy of Sciences (United States) in 1913 and was also a member of the American Philosophical Society, the American Academy of Arts and Sciences, the London Mathematical Society, the French Academy of Sciences and the Union of Czech Mathematicians and Physicists. The American Association for the Advancement of Science decided to set up a prize for the most major contribution to the advancement of science. Dickson was the first recipient of the prize, being awarded $1,000 in 1924 for his work on the arithmetics of algebras. He was also the first recipient of the Cole Prize for algebra awarded by the American Mathematical Society in 1928 for his book Algebren und ihre Zahlentheorie published in Zürich and Leipzig in 1927. Dickson was much involved with the American Mathematical Society, becoming its president in 1917-1918 having earlier, in 1913, been its Colloquium Lecturer. He gave his presidential address in December 1918 on the topic Mathematics in War Perspective in which he criticised the United States for falling short of making the mathematical preparations of Britain, France, and Germany. He said:-

Let it not again become possible that thousands of young men shall be so seriously handicapped in their army and navy work by lack of adequate preparation in mathematics.

As a final comment on honours given to Dickson, we note that Princeton (1941) and Harvard (1936) were among the universities that awarded him honorary degrees.

We mentioned at the beginning of this article that Dickson always considered himself a Texan, so it was natural that he should return 'home' after he retired. His level of research activity throughout his career had been higher than most mathematicians could imagine, and perhaps because of this he effectively gave up research during the fifteen years of his retirement.

As to Dickson's character he is described in [22] as follows:-

A hard-bitten character, Dickson tended to speak his mind bluntly; he was always sparing in his praise for the work of others. ... he indulged his serious passions for bridge and billiards and reportedly did not like to lose at either game.

He certainly had a major impact on mathematics in America and it was in large part due to him that America's role in research into algebra was transformed.