Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an Americanmathematician. He was one of the first American researchers inabstract algebra, in particular the theory of finitefields andclassical groups, and is also remembered for a three-volume history ofnumber theory,History of the Theory of Numbers. The L. E. Dickson instructorships at theUniversity of Chicago Department of Mathematics are named after him.

Dickson considered himself a Texan by virtue of having grown up inCleburne, where his father was a banker, merchant, and real estate investor. He attended theUniversity of Texas at Austin, whereGeorge Bruce Halsted encouraged his study of mathematics. Dickson earned a B.S. in 1893 and an M.S. in 1894, under Halsted's supervision. Dickson first specialised in Halsted's own specialty,geometry.^[2]

Both theUniversity of Chicago andHarvard University welcomed Dickson as a Ph.D. student, and Dickson initially accepted Harvard's offer, but chose to attend Chicago instead. In 1896, when he was only 22 years of age, he was awarded Chicago's first doctorate in mathematics, for a dissertation titledThe Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group, supervised byE. H. Moore.

Dickson then went toLeipzig andParis to study underSophus Lie andCamille Jordan, respectively. On returning to the US, he became an instructor at theUniversity of California. In 1899 and at the extraordinarily young age of 25, Dickson was appointed associate professor at the University of Texas. Chicago countered by offering him a position in 1900, and he spent the balance of his career there. At Chicago, he supervised 53 Ph.D. theses; his most accomplished student was probablyA. A. Albert. He was a visiting professor at theUniversity of California in 1914, 1918, and 1922. In 1939, he returned to Texas to retire.

Dickson married Susan McLeod Davis in 1902; they had two children, Campbell and Eleanor.

Dickson was elected to theNational Academy of Sciences in 1913, and was also a member of the American Philosophical Society, theAmerican Academy of Arts and Sciences, theLondon Mathematical Society, theFrench Academy of Sciences and the Union of Czech Mathematicians and Physicists. Dickson was the first recipient of a prize created in 1924 by The American Association for the Advancement of Science, for his work on the arithmetics of algebras. Harvard (1936) and Princeton (1941) awarded him honorary doctorates.

Dickson presided over theAmerican Mathematical Society in 1917–1918. His December 1918 presidential address, titled "Mathematics in War Perspective", criticized American mathematics for falling short of those of Britain, France, and Germany:

"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."

In 1928, he was also the first recipient of theCole Prize for algebra, awarded annually by the AMS, for his bookAlgebren und ihre Zahlentheorie.

It appears that Dickson was a hard man:

"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."^[3]

"He delivered terse and unpolished lectures and spoke sternly to his students. ... 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."^[4]

"Dickson had a sudden death trial for his prospective 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."^[4]

Dickson had a major impact on American mathematics, especiallyabstract algebra. His mathematical output consists of 18 books and more than 250 papers. TheCollected Mathematical Papers of Leonard Eugene Dickson fill six large volumes.

In 1901, Dickson published his first bookLinear groups with an exposition of the Galois field theory, a revision and expansion of his Ph.D. thesis. Teubner in Leipzig published the book, as there was no well-established American scientific publisher at the time. Dickson had already published 43 research papers in the preceding five years; all but seven onfinite linear groups. Parshall (1991) described the book as follows:

"Dickson presented a unified, complete, and general theory of theclassical linear groups—not merely over theprime field GF(p) asJordan had done—but over the generalfinite field GF(pⁿ), and he did this against the backdrop of a well-developed theory of these underlyingfields. ... his book represented the first systematic treatment offinite fields in the mathematical literature."

An appendix in this book lists the non-abelian simple groups then known having order less than 1 billion. He listed 53 of the 56 having order less than 1 million. The remaining three were found in 1960, 1965, and 1967.

Dickson worked onfinite fields and extended the theory oflinear associative algebras initiated byJoseph Wedderburn andCartan.

He started the study ofmodular invariants of a group.

In 1905, Wedderburn, then at Chicago on a Carnegie Fellowship, published a paper that included three claimed proofs of a theorem stating that all finitedivision algebras werecommutative, now known asWedderburn's theorem. The proofs all made clever use of the interplay between theadditive group of a finitedivision algebraA, and themultiplicative groupA* = A − {0}.Karen Parshall noted that the first of these three proofs had a gap not noticed at the time. Dickson also found a proof of this result but, believing Wedderburn's first proof to be correct, Dickson acknowledged Wedderburn's priority. But Dickson also noted that Wedderburn constructed his second and third proofs only after having seen Dickson's proof. She concluded that Dickson should be credited with the first correct proof.^[5]

Dickson's search for a counterexample to Wedderburn's theorem led him to investigatenonassociative algebras, and in a series of papers he found all possible three and four-dimensional (nonassociative)division algebras over afield.

In 1919 Dickson constructedCayley numbers by a doubling process starting withquaternions${\displaystyle \mathbb{H}}$.^[6] His method was extended to a doubling of${\displaystyle \mathbb{R}}$ to produce${\displaystyle \mathbb{C}}$, and of${\displaystyle \mathbb{C}}$ to produce${\displaystyle \mathbb{H}}$ byA. A. Albert in 1922, and the procedure is known now as theCayley–Dickson construction ofcomposition algebras.

Dickson proved many interesting results innumber theory, using results ofVinogradov to deduce the idealWaring theorem in his investigations ofadditive number theory. He proved theWaring's problem for${\displaystyle k\ge 7}$ under the further condition of

${\displaystyle (3^k+1)/(2^k-1)\le [{1.5}^k]+1}$

independently ofSubbayya Sivasankaranarayana Pillai who proved it for${\displaystyle k\ge 6}$ ahead of him.^[7]

The three-volumeHistory of the Theory of Numbers (1919–23) is still much consulted today, covering divisibility and primality,Diophantine analysis, andquadratic and higher forms. The work contains little interpretation and makes no attempt to contextualize the results being described, yet it contains essentially every significant number theoretic idea from the dawn of mathematics up to the 1920s except for quadratic reciprocity and higher reciprocity laws. A planned fourth volume on these topics was never written.A. A. Albert remarked that this three volume work "would be a life's work by itself for a more ordinary man."

• Dickson, Leonard Eugene (1958) [1901],Magnus, Wilhelm (ed.),Linear groups: With an exposition of the Galois field theory, Dover Phoenix editions, New York:Dover Publications,ISBN 978-0-486-49548-4,MR 0104735^[8][9] (online hathitrust. B.G. Teubner's Sammlung von lehrbüchern auf dem gebiete der mathematischen wissenschaften mit einschluss ihrer anwendungen. Bd. VI. B. G. Teubner. 1901.)
• Dickson, Leonard Eugene (1903) [1903],Introduction to the theory of algebraic equations, New York:John Wiley & Sons^[10] (online hathitrust. J. Wiley & Sons. 1903.)
• Dickson, Leonard Eugene (2010) [1914],Linear algebras, Reprinting of Cambridge Tracts in Mathematics and Mathematical Physics, No. 16, Naub press,ISBN 978-1-178-36805-5,MR 0118745^[11]
• Dickson, Leonard Eugene (2010) [1914],Algebraic invariants, Cornell University Library,ISBN 978-1-4297-0042-9^[12]
• Dickson, Leonard Eugene (2004) [1914],On invariants and the theory of numbers, Dover Phoenix editions, New York:Dover Publications,ISBN 978-0-486-43828-3,MR 0201389^[13]
• Dickson, Leonard Eugene (2009) [1914],Elementary theory of equations, Cornell University Library,ISBN 978-1-112-27988-1
• Dickson, Leonard Eugene (2005) [1919],History of the theory of numbers. Vol. I: Divisibility and primality., New York:Dover Publications,ISBN 978-0-486-44232-7,MR 0245499^[14]
• Dickson, Leonard Eugene (2005) [1920],History of the theory of numbers. Vol. II: Diophantine analysis, New York:Dover Publications,ISBN 978-0-486-44233-4,MR 0245500^[15]
• Dickson, Leonard Eugene (2005) [1923],History of the theory of numbers. Vol. III: Quadratic and higher forms, New York:Dover Publications,ISBN 978-0-486-44234-1,MR 0245501
• Dickson, Leonard Eugene (2009) [1922],First course in the theory of equations, University of Michigan Library
• Dickson, Leonard Eugene (1960) [1923],Algebras and their arithmetics, New York:Dover Publications,ISBN 978-0-486-60616-3,MR 0111764^[16]
• 1926.Modern algebraic theories^[17]
• 1923, 1928.Algebraic Numbers. Report with others for U. S. National Research Council.
• 1929.Introduction to the Theory of Numbers^[18]
• 1930.Studies in the Theory of Numbers^[19]
• 1935. (withG. A. Bliss) "Biographical Memoir of Eliakim Hastings Moore 1862–1932."
• 1935.Researches on Waring's problem
• 1938. (withHans Frederick Blichfeldt andGeorge Abram Miller)Theory and Applications of Finite Groups
• 1938.Algebras And Their Arithmetics (1st edn. in 1923)
• 1939.Modern Elementary Theory of Numbers
• 1939.New First Course in the Theory of Equations
• Plane Trigonometry With Practical Applications
• Dickson, Leonard Eugene (1975), Albert, A. Adrian (ed.),The collected mathematical papers of Leonard Eugene Dickson, vol. I–VI, New York: AMS Chelsea Publishing,ISBN 978-0-8284-0273-6,MR 0749229MR0441665

• O'Connor, John J.;Robertson, Edmund F.,"Leonard Eugene Dickson",MacTutor History of Mathematics Archive,University of St Andrews
• Leonard Eugene Dickson at theMathematics Genealogy Project
• Works by Leonard Eugene Dickson atProject Gutenberg
• Works by or about Leonard Eugene Dickson at theInternet Archive

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