**Leonard Dickson**, or **L E Dickson** as he 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(p)^{n}, 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, is 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 contained 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 Zurich 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 criticise 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 an 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.

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

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