#### by A. A. Albert

Leonard Eugene Dickson was born in Independence, Iowa, on January 22, 1874. He was a brilliant undergraduate at the University of Texas, receiving his B.S. degree as valedictorian of his class in 1893. He was a chemist with the Texas Biological Survey from 1892 to 1893. He served as a teaching fellow at the University of Texas, receiving the M.A. degree in 1894. He held a fellowship at the University of Chicago from 1894 to 1896, and was awarded its first Ph.D. in mathematics in 1896. He spent the year 1896–1897 in Leipzig and Paris, was instructor in mathematics at the University of California 1897–1899, associate professor at Texas 1899–1900, assistant professor at Chicago 1900–1907, associate professor 1907–1910, and professor in 1910. He was appointed to the Eliakim Hastings Moore Distinguished Professorship in 1928, and became professor emeritus in 1939. He served as visiting professor at the University of California in 1914, 1918, and 1922.

Professor Dickson was awarded the $1,000 A.A.A.S. Prize
in 1924 for his work on the arithmetics of algebras. He was awarded
the Cole Prize of the American Mathematical
Society in 1928 for his book
*Albegren und ihre Zahlentheorie*
[42].
He served as editor of the *Monthly* 1902–1908, and the
*Transactions* from 1911 to 1916, and he was president of the American Mathematical
Society from 1916–1918. He was elected to membership in the National
Academy of Sciences in 1913, and was a member of the American
Philosophical Society, the American Academy of Arts and Sciences, and
the London Mathematical Society. He was also a foreign member of the
Academy of the Institute of France, and an honorary member of the
Czechoslovakian Union of Mathematics and Physics. He was awarded the
honorary Sc.D. degree by Harvard in 1936 and Princeton in 1941.

Professor Dickson died in Texas on January 17, 1954.

Dickson was one of our most prolific mathematicians. His bibliography
(prepared by
Mr. Richard Block,
a student at the University of
Chicago) contains 285 titles. Of these, eighteen are books, one a joint
book with
Miller
and
Blichfeldt
[33].
One of the books is his major three-volume
*History of the theory of numbers*
[35],
[36],
[39],
which would be a life’s work by itself for a more ordinary man.

Dickson was an inspiring teacher. He supervised the doctorate dissertations of at least fifty-five Chicago Ph.D.s. He helped his students to get started in research after the Ph.D. and his books had a world-wide influence in stimulating research.

We now pass on to a brief discussion of Dickson’s research.

#### 1. Linear groups

Dickson’s first major research effort was a study of
finite linear groups. All but seven of his first forty-three papers
were on that subject, and this portion of his work led to his famous
first book
[3].
The linear groups which had been investigated by
Galois,
Jordan,
and
Serret
were all groups over the fields of __\( p \)__ elements.
Dickson generalized their results to linear groups over an
arbitrary finite field. He obtained many new systems of simple groups,
and he closed his book with a still valuable summary of the known
systems of simple groups.

Dickson’s work on linear groups continued until 1908, and he wrote about forty-four additional papers on the subject. In these later papers he studied the isomorphism of certain simple groups and questions about the existence of certain types of subgroups. He also derived a number of theorems on infinite linear groups.

#### 2. Finite fields and Chevalley’s theorem

In
[3].
Dickson gave the first extensive exposition of the theory of finite fields.
He applied
his deep knowledge of that subject not only to linear groups but to
other problems which we shall discuss later. He studied irreducibility
questions over a finite field in
[11],
the Galois theory in
[10],
and forms whose values are squares in
[19].
His knowledge of the role
of the non-null form was shown in
[20].
In
[16]
Dickson made the
following statement: “For a finite field it seems to be true that
every form of degree __\( m \)__ in __\( m + 1 \)__ variables vanishes for values not all
zero in the field.”
This result was first proved by
C. Chevalley
in his paper
*Demonstration d’une hypothese de M. Artin*,
Hamb. Abh, vol. 11 (1935), pp. 73–75.
At least the conjecture should have been
attributed to Dickson, who actually proved the theorem for __\( m = 2, 3 \)__.

#### 3. Invariants

Several of Dickson’s early papers were concerned with the problems of the algebraic geometry of his time. For example, see [1], [5], [4]. This work led naturally to his study of algebraic invariants, and his interest in finite fields to modular invariants. He wrote a basic paper on the latter subject in [18], and many other papers on the subject. In these papers he devoted a great deal of space to the details of a number of special cases. His book [24], on the classical theory of algebraic invariants, was published in 1914, the year after the appearance of his colloquium lectures. His amazing productivity is attested to by the fact that he also published his book [25] on linear algebras in 1914.

#### 4. Algebras

Dickson played a major role in research on linear
algebras. He began with a study of finite division algebras in
[6],
[7],
[9]
and
[8].
In these papers he determined all three- and
four-dimensional (nonassociative) division algebras over a field of
characteristic not two, a set of algebras of dimension six, and a
method for constructing algebras of dimension __\( mk \)__ with a subfield of
the dimension __\( m \)__. In
[15]
he related the theory of ternary cubic forms
to the theory of three-dimensional division algebras. His last paper
on non-associative algebras
[47]
appeared in 1937 and contained
basic results on algebras of degree two.

Reference has already been made to Dickson’s first book on linear algebras. In that text he gave a proof of his result that a real Cayley division algebra is actually a division algebra. He presented the Cartan theory of linear associative algebras rather than the Wedderburn theory, but stated the results of the latter theory in his closing chapter without proofs. The present value of this book is enhanced by numerous bibliographical references.

Dickson defined cyclic algebras in a
*Bulletin* abstract of vol. 12
(1905–1906). His paper
[21]
on the subject did not appear until
1912, where he presented a study of algebras of dimension 16.

Dickson’s work on the arithmetics of algebras first appeared in [37]. His major work on the subject of arithmetics was presented in [38] where he also gave an exposition of the Wedderburn theory. See also [44] and [45].

The text [42] is a German version of [38]. However, the new version also contains the results on crossed product algebras which had been published in [46], and contains many other items of importance.

#### 5. Theory of numbers

Dickson always said that mathematics is the
queen of the sciences, and that the theory of numbers is the queen of
mathematics. He also stated that he had always wished to work in the
theory of numbers and that he wrote his monumental three-volume
*History of the theory of numbers*
[35],
[36],
[39]
so that he could know all of the
work which had been done in the subject. His first paper
[2]
contained a generalization of the elementary Fermat theorem which
arose in connection with finite-field theory. He was interested in the
existence of perfect numbers, and wrote
[23]
and
[22]
on the related
topic of abundant numbers. His interest in Fermat’s last theorem
appears in
[34],
[12],
[14],
[13]
and
[17].
During 1926–1930 he
spent most of his energy on research in the arithmetic theory of
quadratic forms, in particular on universal forms.

Dickson’s interest in additive number theory began in 1927 with
[43].
He wrote many papers on the subject during the remainder of his
life. The analytic results of
Vinogradov
gave Dickson the hope of
proving the so-called ideal
Waring
theorem. This he did in a long
series of papers. His final result is an almost complete verification
of the conjecture made by
J. A. Euler
in 1772. That conjecture stated
that every positive integer is a sum of __\( J \)__ __\( n \)__-th powers, where we
write __\( 3^n = 2^n q + r \)__, __\( \,J = 2^n > r > 0 \)__, and __\( J = 2^n + q - 2 \)__.
Dickson showed that if __\( n > 6 \)__ this value is correct unless
__\( q+ r+3 > 2^n \)__. It is still not known whether or not this last inequality
is possible, but if it does occur the number __\( g(n) \)__ of such __\( n \)__-th
powers required to represent all integers is __\( J + f \)__ or __\( J+f-1 \)__,
according as __\( fq + f+ q = 2^n \)__ or __\( fq + f+q > 2^n \)__, where __\( f \)__ is the
greatest integer in __\( (4/3)^n \)__ .

#### 6. Miscellaneous

We close by mentioning Dickson’s interest in the
theory of matrices which is best illustrated by his text,
*Modern algebraic theories*
[41].
His geometric work in
[26],
[30],
[29],
[32],
[27],
[31]
and
[28]
must also be mentioned, as well as his
interesting monograph
[40]
on differential equations from the Lie-group standpoint.