Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
. He was one of the first American researchers in
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, in particular the theory of finite
fields and
classical group
In mathematics, the classical groups are defined as the special linear groups over the reals \mathbb, the complex numbers \mathbb and the quaternions \mathbb together with special automorphism groups of Bilinear form#Symmetric, skew-symmetric an ...
s, and is also remembered for a three-volume history of
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, ''
History of the Theory of Numbers''. The L. E. Dickson instructorships at the
University of Chicago
The University of Chicago (UChicago, Chicago, or UChi) is a Private university, private research university in Chicago, Illinois, United States. Its main campus is in the Hyde Park, Chicago, Hyde Park neighborhood on Chicago's South Side, Chic ...
Department of Mathematics are named after him.
Life
Dickson considered himself a Texan by virtue of having grown up in
Cleburne, where his father was a banker, merchant, and real estate investor. He attended the
University of Texas at Austin
The University of Texas at Austin (UT Austin, UT, or Texas) is a public university, public research university in Austin, Texas, United States. Founded in 1883, it is the flagship institution of the University of Texas System. With 53,082 stud ...
, where
George 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
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
.
[ A. A. Albert (1955]
Leonard Eugene Dickson 1874–1954
from National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...
Both the
University of Chicago
The University of Chicago (UChicago, Chicago, or UChi) is a Private university, private research university in Chicago, Illinois, United States. Its main campus is in the Hyde Park, Chicago, Hyde Park neighborhood on Chicago's South Side, Chic ...
and
Harvard University
Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...
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 titled ''The Analytic Representation of Substitutions on a Power of a Prime Number of Letters with a Discussion of the Linear Group'', supervised by
E. H. Moore.
Dickson then went to
Leipzig
Leipzig (, ; ; Upper Saxon: ; ) is the most populous city in the States of Germany, German state of Saxony. The city has a population of 628,718 inhabitants as of 2023. It is the List of cities in Germany by population, eighth-largest city in Ge ...
and
Paris
Paris () is the Capital city, capital and List of communes in France with over 20,000 inhabitants, largest city of France. With an estimated population of 2,048,472 residents in January 2025 in an area of more than , Paris is the List of ci ...
to study under
Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. He also made substantial cont ...
and
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...
, respectively. On returning to the US, he became an instructor at the
University of California
The University of California (UC) is a public university, public Land-grant university, land-grant research university, research university system in the U.S. state of California. Headquartered in Oakland, California, Oakland, the system is co ...
. 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 probably
A. A. Albert. He was a visiting professor at the
University of California
The University of California (UC) is a public university, public Land-grant university, land-grant research university, research university system in the U.S. state of California. Headquartered in Oakland, California, Oakland, the system is co ...
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 the
National Academy of Sciences
The National Academy of Sciences (NAS) is a United States nonprofit, NGO, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the ...
in 1913, and was also a member of the American Philosophical Society, the
American Academy of Arts and Sciences
The American Academy of Arts and Sciences (The Academy) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and other ...
, the
London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...
, the
French Academy of Sciences
The French Academy of Sciences (, ) is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific method, scientific research. It was at the forefron ...
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 the
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
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 the
Cole Prize
The Frank Nelson Cole Prize, or Cole Prize for short, is one of twenty-two prizes awarded to mathematicians by the American Mathematical Society, one for an outstanding contribution to algebra, and the other for an outstanding contribution to numbe ...
for algebra, awarded annually by the AMS, for his book ''Algebren 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."
:"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."
:"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."
[
]
Work
Dickson had a major impact on American mathematics, especially abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
. His mathematical output consists of 18 books and more than 250 papers. The ''Collected Mathematical Papers of Leonard Eugene Dickson'' fill six large volumes.
The algebraist
In 1901, Dickson published his first book ''Linear 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 on finite linear groups. Parshall (1991) described the book as follows:
:"Dickson presented a unified, complete, and general theory of the classical linear groups—not merely over the prime field
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is wid ...
GF(''p'') as Jordan
Jordan, officially the Hashemite Kingdom of Jordan, is a country in the Southern Levant region of West Asia. Jordan is bordered by Syria to the north, Iraq to the east, Saudi Arabia to the south, and Israel and the occupied Palestinian ter ...
had done—but over the general finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
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 field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s 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 on finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
s and extended the theory of linear associative algebras initiated by Joseph Wedderburn and Cartan.
He started the study of modular 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 finite division algebras were commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
, now known as Wedderburn's theorem. The proofs all made clever use of the interplay between the additive group of a finite division algebra ''A'', and the multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
*the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
''A''* = ''A'' − . 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.
Dickson's search for a counterexample to Wedderburn's theorem led him to investigate nonassociative algebras, and in a series of papers he found all possible three and four-dimensional (nonassociative) division algebras over a field.
In 1919 Dickson constructed Cayley numbers by a doubling process starting with quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s . His method was extended to a doubling of to produce , and of to produce by A. A. Albert in 1922, and the procedure is known now as the Cayley–Dickson construction
In mathematics, the Cayley–Dickson construction, sometimes also known as the Cayley–Dickson process or the Cayley–Dickson procedure produces a sequence of algebra over a field, algebras over the field (mathematics), field of real numbers, eac ...
of composition algebras.
The number theorist
Dickson proved many interesting results in number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, using results of Vinogradov to deduce the ideal Waring theorem in his investigations of additive number theory
Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigro ...
. He proved the Waring's problem for under the further condition of
:
independently of Subbayya Sivasankaranarayana Pillai who proved it for ahead of him.
The three-volume '' History of the Theory of Numbers'' (1919–23) is still much consulted today, covering divisibility and primality, Diophantine analysis, and quadratic 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."
Bibliography
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* 1926. ''Modern algebraic theories''
*1923, 1928. ''Algebraic Numbers''. Report with others for U. S. National Research Council.
*1929. ''Introduction to the Theory of Numbers''
*1930. ''Studies in the Theory of Numbers''
*1935. (with G. A. Bliss) "Biographical Memoir of Eliakim Hastings Moore 1862–1932."
*1935. ''Researches on Waring's problem''
*1938. (with Hans Frederick Blichfeldt and George 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''
*
Notes
External links
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{{DEFAULTSORT:Dickson, Leonard Eugene
1874 births
1954 deaths
19th-century American mathematicians
20th-century American mathematicians
American algebraists
American number theorists
University of Texas at Austin College of Natural Sciences alumni
University of Chicago alumni
University of Texas at Austin faculty
University of Chicago faculty
American historians of mathematics
Presidents of the American Mathematical Society
Members of the United States National Academy of Sciences
People from Independence, Iowa
People from Cleburne, Texas
Mathematicians from Iowa
The American Mathematical Monthly editors