#### by Nathan Jacobson

Adrian Albert, one of the foremost algebraists of the world and President of the American Mathematical Society from 1965 to 1967, died on June 6, 1972. For almost a year before his death it had become apparent to his friends that his manner had altered from its customary vigor to one which was rather subdued. At first they attributed this to a letdown which might have resulted from Albert’s having recently relinquished a very demanding administrative position (Dean of the Division of Physical Sciences at the University of Chicago) that he had held for a number of years. Eventually it became known that he was gravely ill of physical causes that had their origin in diabetes with which he had been afflicted for many years.

Albert was a first generation American and a second generation American mathematician following that of E. H. Moore, Oswald Veblen, L. E. Dickson and G. D. Birkhoff. His mother came to the United States from Kiev and his father came from England.1 The father had run away from his home in Vilna at the age of fourteen, and on arriving in England, he discarded his family name (which remains unknown) and took in its place the name Albert after the prince consort of Queen Victoria. Albert’s father was something of a scholar, with a deep interest in English literature. He taught school for a while in England but after coming to the United States he became a salesman, a shopkeeper, and a manufacturer. Adrian was born when his father was fifty-five and his mother was thirty-five. It was a second marriage for both parents; his father’s first wife had died in childbirth, and his mother was a widow with two children when she married his father. Adrian was the middle child of a set of three children which his parents had in common. He grew up in a family that was formally orthodox Jewish but not strongly religious. In common with most immigrant families of the period the family had a strong drive toward assimilation and a determination to make the most of the opportunities offered by a comparatively free society undergoing rapid economic expansion with no limits in sight.

Albert spent all of his school years in the Midwest and all but two of these in Chicago. He attended public schools at Chicago and at Iron Mountain, Michigan, and entered the University of Chicago in 1922 where in rapid succession he earned a B.S. degree in 1926, an M.S. degree in 1927, and a Ph.D. in 1928. His advisor for his master’s and his doctoral dissertations was Leonard Eugene Dickson. After his doctorate Albert spent a year at Princeton University as a National Research Council Fellow. He was attracted to Princeton by that great master of associative algebra theory, J. H. M. Wedderburn, who was then a professor at the university. Albert returned to Princeton in 1933, this time as one of the first group of temporary members of the Institute for Advanced Study.

Albert married Frieda Davis in 1927, and they had three children, Alan, Roy, and Nancy, one of whom, Roy, died of diabetes at the age of twenty-three.

Except for two years (1929–1931) as an Instructor at Columbia University and a number of visiting professorships (at Rio de Janeiro, Buenos Aires, University of Southern California, Yale, and the University of California at Los Angeles) all of Albert’s academic career was spent at the University of Chicago. In 1960 he was named Eliakim Hastings Moore Distinguished Service Professor, and he served as Chairman of the Department of Mathematics for three years until he became Dean of the Division of Physical Sciences in 1962. He held this position until 1971 when he reached the mandatory retirement age of sixty-five for the deanship.

Of the mathematicians who influenced Albert most directly we should
list the following: Dickson, who set the direction for almost all of Albert’s
research and whose books, *Algebras and their Arithmetics* (1923) and
*Algebren und ihre Zahlentheorie* (1927), stimulated the great flowering of
associative algebra theory of the 1930’s; Wedderburn, whose elegant
results and methods were an inspiration to Albert;
Hermann Weyl, whose lectures on
Lie groups and especially Lie algebras aroused Albert’s
interest in this subject — an interest which later broadened to encompass
the whole range of nonassociative algebras; and above all,
Solomon Lefschetz, who introduced Albert to the subject of
Riemann matrices
during his postdoctoral year (1928–1929) at Princeton.

Mrs. Albert tells the story of this introduction in a charming fashion. Filling in some mathematical details it runs somewhat as follows. Albert had given a lecture on his dissertation at the Princeton mathematics club. In the audience were Dieudonné, J. H. C. Whitehead and Lefschetz, who had worked on the problem of multiplication algebras of Riemann matrices. Lefschetz apparently sensed that here was a brilliant young algebraist whose interests and power made him ideally suited to attack this problem. After Albert’s talk he described the problem to him. A lively discussion ensued, mostly in the course of wanderings through the streets of Princeton. This lasted for several hours, well past dinnertime, and Mrs. Albert had become quite concerned before Albert finally returned home, apparently in great excitement over his initiation into a fascinating area of classical mathematics which provided a strong motivation for the study of his chosen field of associative algebras.

Lefschetz was certainly right in his judgment. Albert took to the problem on Riemann matrices with great enthusiasm, and as the structure theory of associative algebras was revealed by Albert, Brauer, Hasse and Emmy Noether, Albert could push forward the theory of multiplication algebras of Riemann matrices until he achieved a complete solution of the central problem (which we shall discuss below). For this achievement Albert was awarded the Cole Prize in algebra in 1939.

This was a memorable year for Albert. Besides the Cole Prize award
which he received that year, he was the Colloquium speaker of the
Society for 1939. Moreover, he performed a feat, which we believe has
never been matched, of having the book, *Structure of Algebras*, the
subject of his lectures in print at the same time that the lectures were
delivered.

Around 1942 Albert’s research interests shifted from associative to nonassociative algebras. He wrote many important papers in this field (which we shall discuss below). In 1965 Albert returned to his first love, structure theory of associative algebras.

Besides his own important contributions to mathematics, Albert was instrumental in a number of ways in improving the status of the profession. He had a good deal to do with the establishment of government research grants for mathematics on more or less an equal footing with those in the other sciences. He was chairman of the Committee to Prepare a Budget for Mathematics for the National Science Foundation, 1950, and chairman of the Committee on a Survey of Training and Research Potential in the Mathematical Sciences, January 1955–June 1957 (which became known as “The Albert Committee”). He demonstrated that pure mathematicians could be useful in applied and directed research by acting as a consultant for a number of government sponsored research agencies. For a number of years he was associated with the Institute for Defense Analysis as a member of its Board of Trustees and for a year as Director of its Princeton group. He directed the research project SCAMP for several summers and organized and directed the project ALP (known as “Albert’s little project”).

Albert was also a driving force in the creation of the summer research institutes which have become such an important part of the research activities of the Society, supported by the National Science Foundation. He was chairman of the committee which was responsible for the first one of these — on Lie groups and Lie algebras — held at Colby College in Maine in the summer of 1953.

Albert’s role as a “statesman” for mathematics included membership on the Board of Trustees of the Institute for Advanced Study, chairman of the Consultative Committee of the Nice Congress, and Vice-President of the International Mathematical Union.

His influence in mathematics extended also through a large number of gifted students. One of the most distinguished of these, Dan Zelinsky, has written a warm appreciation of Albert as a mathematician and as a person [e37].

Naturally many important honors came his way. He was elected to the National Academy of Sciences in 1943 and was awarded honorary doctorates from Notre Dame, Yeshiva University, and the University of Illinois. He was elected a corresponding member of the Brazilian Academy of Sciences, honorary member of the Argentine Academy of Sciences, and of the Mexican Mathematical Society. He thoroughly enjoyed these honors, but he derived almost as much pleasure from the honors bestowed on fellow algebraists and on his friends. Most of all he enjoyed seeking out a colleague to whom he could communicate his latest discovery, which excited him greatly.

Most of Albert’s important discoveries fall neatly into three categories: I. Associative algebras, II. Riemann matrices, III. Nonassociative algebras. We proceed to give an indication of these and of some interesting isolated results which we shall mention under IV. Miscellaneous.

#### I. Associative algebras

The Wedderburn structure theorems of 1907
on finite dimensional associative algebras over a field focused attention
on the division algebras in this class. In 1906 Dickson had given a construction
of a type of algebra called *cyclic* which included division algebras.
These contain a maximal subfield __\( \mathfrak{Z} \)__ which is cyclic over the base
field __\( \mathfrak{F} \)__, that is, they are
Galois with Galois group __\( G=\langle s\rangle \)__, a
cyclic group generated by a single element __\( s \)__. Moreover, the algebras are generated by
__\( \mathfrak{Z} \)__ and an element __\( u \)__ for which one has the relations
__\begin{equation*}
uz=s(z)u,\qquad z\in\mathfrak{Z},\qquad u^n=\gamma,
\end{equation*}__
where __\( n \)__ is the order of __\( G \)__ and __\( \gamma \)__ is a nonzero element of __\( \mathfrak{F} \)__. The cyclic
algebra, denoted as __\( (\mathfrak{Z}, s, \gamma) \)__, constructed in this way has dimensionality
__\( n^2 \)__ over __\( \mathfrak{F} \)__. In
[e1]
Wedderburn
proved an important sufficient condition
for __\( (\mathfrak{Z}, s,\gamma) \)__ to be a division algebra. He showed that this is the
case if no power of __\( \gamma,\gamma^m \)__ with __\( 0 < m < n \)__, is a norm __\( N_{\mathfrak{Z}/\mathfrak{F}}(z) \)__ of an element
__\( z\in \mathfrak{Z} \)__. Using this criterion it is easy to construct division algebras of any
dimension __\( n^2 \)__.

In 1921 Wedderburn published some other important results on division
algebras
[e2].
Noting that one may as well consider these as algebras
over their centers and so assume that they are *central* in the sense that the
center is the base field __\( \mathfrak{F} \)__, he showed that the dimensionality over this
field is a square, __\( n^2 \)__. More generally, if __\( \mathfrak{U} \)__ is central simple, by one
of Wedderburn’s
structure theorems, __\( \mathfrak{U} \)__ is the algebra __\( M_r(\mathfrak{D}) \)__ of __\( r\times r \)__ matrices
with elements in a central division algebra __\( \mathfrak{D} \)__. Hence if the dimensionality
of __\( \mathfrak{D} \)__ over __\( \mathfrak{F} \)__ is __\( d^2 \)__, then that of __\( \mathfrak{U} \)__ over __\( \mathfrak{F} \)__ is __\( n^2 \)__, where __\( n=dr \)__. Then __\( n \)__ is
called the *degree* of the central simple algebra __\( \mathfrak{U} \)__ and __\( d \)__ is its *index*. In
his 1921 paper, Wedderburn showed also that any maximal subfield __\( \mathfrak{Z} \)__ of a
central division algebra __\( \mathfrak{D} \)__ is a *splitting field*, that is, the algebra __\( \mathfrak{D}^{\mathfrak{Z}}=\mathfrak{Z}\otimes_\mathfrak{F}\mathfrak{D} \)__ is the matrix algebra __\( M_d(\mathfrak{Z}) \)__, and he proved that every central
division algebra of degree three is cyclic. Wedderburn showed also that
Dickson’s cyclic algebras were special cases of a more general type of
algebra which is now called an abelian crossed product. Here the cyclic
field __\( \mathfrak{Z} \)__ is replaced by a Galois extension field of the base field with Galois group an abelian group.

Abelian crossed products were rediscovered by Cecioni [e3], and these were further generalized by Dickson [e4] and [e6] to arbitrary crossed products based on any Galois extension field.

Much of Albert’s early work was concerned with the study of finite
dimensional central simple algebras. His first important result on these was
the theorem, proved in his dissertation
[1],
that every central division
algebra of degree four (dimension sixteen) is a crossed product. This was
the next case to be considered after Wedderburn’s theorem that in degree
three these algebras are cyclic. Albert improved the result in
[2]
by
showing that the degree four central division algebras are crossed products
based on abelian extension fields whose Galois groups are direct products
of two cyclic groups of order two, and he gave a simpler proof of this
result in
[5].
In both of these papers the algebras of characteristic two
were excluded. In a subsequent paper
[10]
he was able to overcome the
difficulties of the characteristic two case. Brauer was the first to show
that the central division algebras of degree four, unlike those of degree
three, need not be cyclic. He constructed an example of such an algebra
which was a tensor product of two (generalized) quaternion algebras
[e7].
Subsequently,
Albert [7]
constructed one which is not such a
product. This was significant in view of another important theorem,
proved by Albert
[35],
stating that a central division algebra __\( \mathfrak{D} \)__ of degree
four is a tensor product of quaternion algebras if and only if
__\( \mathfrak{D}\otimes_\mathfrak{F}\mathfrak{D}\cong M_4(\mathfrak{F}) \)__.

The main goal of the structure theory of algebras of the period 1929–1932 was the determination and classification of finite dimensional
division algebras over the field __\( \boldsymbol{Q} \)__ of rational numbers, or equivalently,
finite dimensional central division algebras over number fields. It was
recognized quite early that this problem had two separate aspects: a
purely algebraic one concerned with properties of algebras valid for all
base fields, and an arithmetic one exploiting the arithmetic of number
fields. Albert recognized the importance of the arithmetic method. However,
he was handicapped in its use by the fact that he was unaware
until rather late of the powerful results of algebraic number theory,
notably, class field theory, which had been developed in Germany. He
did make use of the arithmetic theory of quadratic forms to achieve
definitive results on central division algebras of degree four over number
fields and some important early results on the degree __\( 2^n \)__ case. For example,
he proved that the former are cyclic and are not tensor products of quaternion
algebras, and he proved that the only central division algebras over
number fields which possess involutions, that is, antiautomorphisms of
period two, are the quaternion algebras. This last result was needed for
his study of Riemann matrices which we shall discuss below.

Albert’s main contributions were on the purely algebraic side. There is
a substantial overlap between his results on central simple algebras and
those of the German school of algebraists of the period of the early thirties,
especially those of Richard Brauer and of Emmy Noether. Albert obtained
independently all the algebraic results on splitting fields, extensions
of isomorphisms and tensor products which were needed to obtain
the fundamental theorems on division algebras over number fields.
Of central importance for the algebraic theory is the group of classes of
central simple algebras which was introduced by Brauer in 1929
[e7].
We recall the definition. Two (finite dimensional) central simple algebras
__\( \mathfrak{U} \)__ and __\( \mathfrak{B} \)__ over a field __\( \mathfrak{F} \)__ are said to be
*similar* (__\( \sim \)__) if there exist positive
integers __\( m \)__ and __\( n \)__ such that the matrix algebras __\( M_m(\mathfrak{U}) \)__ and __\( M_n(\mathfrak{B}) \)__ are
isomorphic. This is an equivalence relation. Denoting the similarity
class of __\( \mathfrak{U} \)__ as __\( \{\mathfrak{U}\} \)__, one defines a product of such classes
by __\( \{\mathfrak{U}\}\{\mathfrak{B}\}=\{\mathfrak{U}\otimes_\mathfrak{F}\mathfrak{B}\} \)__.
This gives a commutative group __\( B(\mathfrak{F}) \)__ called *the Brauer
group of the field* __\( \mathfrak{F} \)__. The unit of the group is the set of matrix algebras
__\( M_n(\mathfrak{F}) \)__, __\( n=1,2,\dots \)__, and the inverse of __\( \{\mathfrak{U}\} \)__ is __\( \{\mathfrak{U}^{\mathrm{op}}\} \)__, where __\( \mathfrak{U}^{\mathrm{op}} \)__ is the
opposite algebra of __\( \mathfrak{U} \)__. In a beautiful paper
[3]
published in 1931,
Albert
essentially rediscovered the Brauer group. In this paper he proved Brauer’s
main theorem that __\( B(\mathfrak{F}) \)__ is a torsion group; more precisely, if __\( \mathfrak{U} \)__ has
index __\( m \)__, that is, if the degree of the division algebra __\( \mathfrak{D} \)__ in __\( \{\mathfrak{U}\} \)__ is __\( m \)__, then
__\( \{\mathfrak{U}\}^m=1 \)__. Moreover, if __\( e \)__ is the order of __\( \{\mathfrak{U}\} \)__ in __\( B(\mathfrak{F}) \)__, then __\( e \)__ and __\( m \)__ have the
same prime factors. The integer __\( e \)__ is called the *exponent* of __\( \mathfrak{U} \)__. Albert’s
proofs are based on Wedderburn’s norm condition for cyclic algebras
to be division algebras and theorems reducing considerations to the cyclic
case; for example, if __\( \mathfrak{D} \)__ is a central division algebra of prime degree __\( p \)__,
then there exists an extension field __\( \mathfrak{R} \)__ of the base field of dimensionality
prime to __\( p \)__ such that __\( \mathfrak{D}^\mathfrak{R}=\mathfrak{R}\otimes_\mathfrak{F}\mathfrak{D} \)__ is a cyclic division algebra over __\( \mathfrak{R} \)__.
Another key tool in Albert’s method was the following theorem which he
called the *index reduction factor theorem*: Let __\( \mathfrak{D} \)__ be a central division algebra
of degree __\( d \)__ and __\( \mathfrak{R} \)__ an extension field of the base field __\( \mathfrak{F} \)__ with dimensionality __\( r \)__. Then __\( \mathfrak{D}^\mathfrak{R}=M_q(\mathfrak{E}) \)__, where __\( \mathfrak{E} \)__ is a central division algebra over __\( \mathfrak{R} \)__ and __\( q \)__ is a divisor of __\( d \)__ and __\( r \)__. Albert’s primary interest in the theorem that
__\( \{\mathfrak{U}\}^m = 1 \)__ was its consequence that any central division algebra is a tensor
product of division algebras of prime power degrees which are determined
up to isomorphism. This reduced most questions on these algebras to the
prime power degree case.

The high points of the structure theory of algebras of the 1930’s were
undoubtedly the theorem that every finite dimensional central division
algebra over a number field is cyclic, and the classification of these algebras
by a set of numerical invariants. The latter result amounts to the determination
of the structure of the Brauer group for a number field. Besides
the general theory of central simple algebras we have indicated, the
proofs of these fundamental results required the structure theory of
central simple algebras over __\( p \)__-adic fields due to Hasse, Hasse’s norm
theorem (“the Hasse principle”), and the
Grunwald existence theorem for
certain cyclic extensions of a number field. (Though it was discovered
almost thirty years later by
S. Wang
[e19]
that Grünwald’s formulation
was incorrect, his error did not affect the theorem on algebras. See also
Wang
[e21]
and Hasse
[e22].)
The first proof of the cyclic structure of
central division algebras over number fields was given by Brauer, Hasse
and Noether
[e8].
However, it seemed appropriate that Albert
should share the honor of this achievement, and at Hasse’s suggestion a
joint paper
([4],
1932)
was published by Albert and Hasse giving another
proof of the theorem and the historical background of the problem.

The results which had been obtained up to this point suggested the following
two problems: (I) Is every finite dimensional central division algebra
a crossed product? (II) Is every one of prime degree cyclic? These are
equivalent to the question of existence of a maximal Galois and maximal
cyclic subfield, respectively, for these algebras. The results of Wedderburn
and Albert imply that the answer to the second question is affirmative
for the primes 2 and 3 and for the first for the degrees 2, 3, 4, 6 and 12.
Quite recently
Amitsur showed that the answer to the first question
is negative by showing that for any __\( n \)__ divisible by eight or by the square
of an odd prime there exists a noncrossed product central division algebra
of degree __\( n \)__
[e36].
This leaves intact the second problem, and this is one
on which Albert spent a good deal of effort. It is clear from the definition
that if __\( \mathfrak{U} \)__ is cyclic of degree __\( n \)__, then __\( \mathfrak{U} \)__ contains
an element __\( u \)__ satisfying an irreducible pure equation __\( x^n-\gamma=0 \)__,
__\( \gamma \)__ in the base field __\( \mathfrak{F} \)__. Does the converse
hold? Albert showed this is the case if __\( n=p \)__, a prime
[12],
and is
not the case if __\( n=4 \)__
[22],
For the prime case this reduces the problem (II) to what appears to be a more tractable one: Does every central
division algebra __\( \mathfrak{U} \)__ of prime degree __\( p \)__ contain an element not in __\( \mathfrak{F} \)__ whose
__\( p \)__-th power is in __\( \mathfrak{F} \)__? In 1938 Brauer showed that if __\( \mathfrak{U} \)__ is of degree 5 there
exists a field __\( \mathfrak{R} \)__ containing a tower of fields __\( \mathfrak{F}\subset\mathfrak{R}_1\subset\mathfrak{R}_2\subset\mathfrak{R} \)__ such that the
degree __\( [\mathfrak{R}_1:\mathfrak{F}]=2=[\mathfrak{R}_2:\mathfrak{R}_1] \)__ and
__\( [\mathfrak{R}:\mathfrak{R}_2]=3 \)__, and __\( \mathfrak{U}^\mathfrak{R} \)__ is cyclic
[e16].
This led Albert to consider the following question: Suppose __\( \mathfrak{R} \)__ is a quadratic
extension of __\( \mathfrak{F} \)__ and __\( \mathfrak{U}^\mathfrak{R} \)__ is cyclic of prime degree. Then is __\( \mathfrak{U} \)__ cyclic?
In four papers
appearing between 1965 and 1970,
including his retiring Presidential Address for the Society, Albert considered
this problem for algebras of characteristic __\( p \)__ and degree __\( p \)__
[48],
[50],
[51],
[52].
In
spite of many ingenious arguments and partial results, he was unable to
completely settle this question.

A beautiful chapter in the structure theory of central simple algebras
is the theory of __\( p \)__-algebras which Albert developed in three papers
appearing in 1936 and 1937
[18],
[19],
[20]
(cf. also
[e12]).
These are the central simple algebras of
characteristic __\( p \)__ whose division algebras __\( \mathfrak{D} \)__ in the
Wedderburn theorem (__\( \cong M_n(\mathfrak{D}) \)__) have degree a power of __\( p \)__.
The main results Albert proved about __\( p \)__-algebras are that any such
algebra __\( \mathfrak{U} \)__ is cyclically representable, that is, there exists
an __\( n \)__ such that __\( M_n(\mathfrak{U}) \)__ is a cyclic algebra, and the exponent
of __\( \mathfrak{U} \)__ is the minimum of the exponents of purely inseparable
splitting fields for the algebra.

A generalization of cyclic algebras in which the cyclic maximal subfield
__\( \mathfrak{Z} \)__ is replaced by a separable commutative subalgebra on which a cyclic
group __\( G \)__ acts in such a way that there are no proper subalgebras stabilized
by __\( G \)__ was considered by Albert in
[21]
following earlier work by
Teichmüller in
[e13].
Such *generalized cyclic algebras* arise naturally from cyclic
ones when one extends the base field or forms the tensor powers of a cyclic
algebra.

Most of the important results on associative algebras which Albert
obtained prior to 1939 can be found in an improved form in his AMS
Colloquium book, *Structure of Algebras*. This extremely readable and
beautifully organized book can still be recommended to a beginning student
with a serious interest in structure theory and is an indispensable reference
book for certain aspects of the theory, particularly the theory of __\( p \)__-algebras,
and of algebras with involution.

#### II. Riemann matrices and associative algebras with involution

The theory of multiplications of Riemann matrices has its origin in algebraic
geometry. On a Riemann surface of an algebraic curve of genus __\( p \)__, one
chooses __\( p \)__ linearly independent integrals of the first kind each with __\( 2p \)__
periods __\( \omega_{j\nu} \)__, __\( 1\leqq j\leqq p \)__, __\( 1\leqq \nu \leqq 2p \)__. The __\( p\times 2p \)__
matrix __\( \omega = (\omega_{j\nu}) \)__ of complex
elements satisfies the *Riemann relations*: there exists a __\( 2p\times 2p \)__ nonsingular
skew symmetric matrix __\( C \)__ of rational elements such that __\( \omega\mkern1muC\mkern1mu{}^t\mkern-1mu\omega=0 \)__
(__\( \mkern1mu{}^t\mkern-1mu\omega \)__ the transpose of __\( \omega \)__) and __\( \sqrt{-1}\,\omega\mkern1muC\mkern1mu{}^t\mkern-1mu\bar{\omega} \)__ is
positive definite hermitian. In the theory
of so-called singular correspondences on the Riemann surface, one is led
to consider the multiplications of __\( \omega \)__. These are the __\( 2p\times 2p \)__ rational
matrices __\( A \)__ for which there exists __\( p\times p \)__ complex matrix __\( \alpha \)__ such that
__\( \alpha\omega=\omega A \)__.
The set of these __\( A \)__’s is a finite dimensional algebra of matrices over __\( \boldsymbol{Q} \)__,
the algebra of multiplications of __\( \omega \)__.

Alternatively, the matrices __\( \omega \)__ and their multiplications arise in the theory
of abelian functions, defined to be meromorphic functions of __\( p \)__ complex
variables having a lattice of periods in __\( \boldsymbol{C}^p \)__.

There is another, formally simpler, formulation of Riemann matrices
(the foregoing __\( \omega \)__) and their multiplications due to Weyl
[e10]
which was
suggested by geometric considerations. From the purely formal point of
view one obtains the passage from the classical formulation to Weyl’s by
introducing the __\( 2p\times 2p \)__ matrix
__\begin{equation*}
W=\binom{\omega}{\bar{\omega}}\quad\text{and}\quad
L=\begin{pmatrix}
-\sqrt{-1}\,\,\boldsymbol{1}_p &0\\
0 &\sqrt{-1}\,\,\boldsymbol{1}_p
\end{pmatrix}.
\end{equation*}__
Put __\( R= W^{-1}LW \)__. Then it can be shown that the matrix __\( R \)__ has the following
properties: (1) __\( R \)__ is real, that is __\( R\in M_{2p}(\boldsymbol{R}) \)__; (2) __\( R^2=-\boldsymbol{1}_{2p} \)__;
(3) __\( S=CR \)__ is
positive definite symmetric. Following Weyl, one calls a matrix __\( R\in M_n(\boldsymbol{R}) \)__
(here __\( n=2p \)__) a *Riemann matrix* if __\( R^2=-\boldsymbol{1}_n \)__ and there exists a skew symmetric
matrix __\( C\in M_n(\boldsymbol{Q}) \)__ such that __\( S=CR \)__ is positive definite symmetric.
The matrix __\( C \)__, which is necessarily nonsingular, is called a *principal
matrix* of __\( R \)__. The passage from Weyl’s __\( R \)__ to the classical __\( \omega \)__ can be reversed.
In Weyl’s formulation the *multiplications* appear as the matrices __\( A\in M_n(\boldsymbol{Q}) \)__
commuting with __\( R \)__. The set __\( \mathfrak{U} \)__ of these multiplications is a finite dimensional
algebra over __\( \boldsymbol{Q} \)__ called the *multiplication algebra* of the Riemann matrix __\( R \)__. Weyl observed that for most considerations the condition __\( R^2=-\boldsymbol{1} \)__
__\( (=-\boldsymbol{1}_n) \)__ plays no role. Dropping this, one obtains *generalized Riemann
matrices*. Subsequently Albert
[23],
[17]
considered further generalizations
(including even a characteristic __\( p\neq 0 \)__ situation!). For the sake of simplicity
we shall stick to the case of Riemann matrices in Weyl’s formulation.

The important early work on multiplication algebras is due to
Poincaré, Scorza, Lefschetz and
Rosati. Poincaré achieved a reduction to so-called
*pure* Riemann matrices for which the multiplication algebras are division
algebras. Lefschetz considered the situation in which the multiplication
algebras are commutative. Rosati observed the important fact that if __\( A \)__
is in the multiplication algebra __\( \mathfrak{U} \)__ of a Riemann matrix __\( R \)__ and __\( C \)__ is a
principal matrix, then __\( A^\ast = C^{-1}\mkern1mu{}^t\mkern-1muAC\in\mathfrak{U} \)__. The map __\( A\to A^\ast \)__ is an involution
(antiautomorphism of period two) in __\( \mathfrak{U} \)__. Rosati showed also that if __\( A \)__ is
symmetric under this involution __\( (A^\ast=A) \)__ then its characteristic roots are
real, and if __\( A^\ast =-A \)__ then its characteristic roots are pure
imaginaries
[e5].

The central problem on multiplication algebras of Riemann matrices is
to determine necessary and sufficient conditions that a division algebra
over __\( \boldsymbol{Q} \)__ be the multiplication algebra of a Riemann matrix. For a proof
of sufficiency, one requires a construction of a Riemann matrix whose
multiplication algebra is a given algebra __\( \mathfrak{U} \)__ satisfying the conditions.

Albert’s work on Riemann matrices went hand in hand with the development
of the theory of division algebras. It culminated in the complete
solution of the principal problem, which he published in three papers
appearing in the Annals of Mathematics in 1934 and 1935 (
[14],
[9]
and
[15]).
To achieve this required the development ab initio of the basic
theory of simple algebras with involution. Albert presented improved
versions of this theory in
[17]
and in his *Structure of Algebras*. We shall
indicate first his results on algebras with involution.

We assume throughout that __\( \mathfrak{U} \)__ is finite dimensional simple over a field __\( \mathfrak{F} \)__.
If __\( \mathfrak{U} \)__ has an involution __\( J \)__ (__\( J:a\to a^\ast \)__ such that __\( (a+b)^\ast=a^\ast+b^\ast \)__, __\( (\alpha a)^\ast=\alpha a^\ast \)__
for __\( \alpha\in \mathfrak{F} \)__, __\( (ab)^\ast=b^\ast a^\ast \)__), then the center __\( \mathfrak{C} \)__ of __\( \mathfrak{U} \)__ is stabilized by __\( J \)__ and the
restriction of __\( J \)__ to __\( \mathfrak{C} \)__ is either the identity map or an automorphism of
period two. Accordingly, the involution is of *first kind* or *second kind*.
Albert showed that __\( \mathfrak{U} \)__ has an involution if and only if for any __\( m=1, 2,\dots \)__, the matrix algebra __\( M_m(\mathfrak{U}) \)__ has an involution having the same effect
on the center (which can be identified with the center of the matrix algebra).
He showed also that if __\( \mathfrak{U} \)__ has an involution __\( J \)__ and __\( x \)__ is an element of
__\( \mathfrak{U} \)__ whose minimum polynomial over the center __\( \mathfrak{C} \)__ is irreducible and has
coefficients that are __\( J \)__-symmetric, then __\( \mathfrak{U} \)__ has an involution __\( T \)__ leaving __\( x \)__
fixed and having the same effect on __\( \mathfrak{C} \)__ as __\( J \)__. Assuming __\( \mathfrak{U} \)__ is of dimension __\( n^2 \)__
over its center __\( \mathfrak{C} \)__ and contains a subfield of the form __\( \mathfrak{X} \otimes_\mathfrak{F}\mathfrak{C} \)__, where __\( \mathfrak{X} \)__
is __\( n \)__-dimensional Galois over __\( \mathfrak{F} \)__, and __\( \mathfrak{C} \)__ is either __\( \mathfrak{F} \)__ or a separable quadratic
extension of __\( \mathfrak{F} \)__, he gave a necessary and sufficient condition in terms of a
factor set for __\( \mathfrak{U} \)__ to have an involution. This was used to give constructions
which in principle yield all simple algebras with involution. Albert also
used these results to prove that a central simple algebra has an involution
if and only if it has exponent one or two in the Brauer group. One can
combine this with one of the results of I to conclude that a central
division algebra of degree four has an involution if and only if it is a tensor
product of quaternion algebras. It seems unlikely that this is true for degree
greater than four but we believe that this remains an open question.

Albert proved that if a division algebra over a number field has an involution
of first kind, then the algebra is a quaternion algebra over its
center. Moreover, he determined the division algebras over number fields
having involutions of second kind. He showed that any such algebra is
cyclic __\( (\mathfrak{Z}, s,\gamma) \)__ over its center __\( \mathfrak{C} \)__, where the cyclic field __\( \mathfrak{Z} \)__
over __\( \mathfrak{C} \)__ has the
form __\( \mathfrak{Z}_0\otimes_{\mathfrak{C}_0}\mathfrak{C} \)__, __\( \mathfrak{C}_0 \)__ the subfield of symmetric
elements of __\( \mathfrak{C} \)__, __\( \mathfrak{Z}_0 \)__ cyclic
over __\( \mathfrak{C}_0 \)__, and __\( \mathfrak{C} \)__ separable quadratic over __\( \mathfrak{C}_0 \)__. Moreover,
__\( \gamma\in\mathfrak{C} \)__, and if
__\( \bar{\gamma} \)__ is its conjugate in __\( \mathfrak{C} \)__ over __\( \mathfrak{C}_0 \)__, then __\( \gamma\bar{\gamma} \)__
is a norm of an element of __\( \mathfrak{Z}_0 \)__.
Conversely any division algebra having the indicated cyclic structure does
have an involution of second kind. In a later paper
[47],
Albert obtained
a similar result for division algebras of degree three (over their centers)
for arbitrary base fields.

These results, especially those on division algebras with involutions over
number fields, provided the machinery for the solution of the problem of
multiplication algebras for Riemann matrices. Using Rosati’s theorem,
one sees that the center __\( \mathfrak{C} \)__ of such an algebra is totally real if the (Rosati)
involution is of first kind and is a pure imaginary quadratic extension
of a totally real field if the involution is of second kind. Besides these
conditions on __\( \mathfrak{C} \)__, there are supplementary conditions on the quaternion
division algebras and the cyclic algebras, which occurred in our description
of the division algebras with involution over number fields, that must
be fulfilled for these to be multiplication algebras of Riemann matrices.
In Albert’s proof of the sufficiency of the conditions he had derived, he
made use of the
Hilbert irreducibility theorem for number fields.

In an exposition [e30] of the theory of Riemann matrices, C. L. Siegel made some notable improvements on Albert’s results. We should mention also that Weyl in [e11] gave an alternative treatment of the subject based on Brauer factor sets.

#### III. Nonassociative algebras

From about 1942 to 1965, when he returned to the problem of existence of noncyclic associative division algebras of prime degree, most of Albert’s research was in the area of nonassociative algebra: structure theory of nonassociative algebras, quasigroups, nonassociative division rings, and nondesarguesian projective planes. In our account of his contributions to this rather broad field of mathematics, we shall be selective, picking out what we consider his most important work — judged from the criterion of general mathematical interest. From this point of view, Albert’s discoveries on Jordan algebras are undoubtedly his most important ones in nonassociative algebra, and these are perhaps on a par with his work on associative algebras and Riemann matrices. We shall begin our account with this work, and we shall first sketch the story of Jordan algebras before Albert took them up as a subject of intensive study.

The study of the class of algebras which now bear his name was initiated
in 1932 by the physicist, P. Jordan. His declared objective was to
achieve a better formalism for quantum mechanics than one based on
selfadjoint operators in Hilbert space. Observing that the set of these
operators is a vector space over __\( \boldsymbol{R} \)__ which is closed under the product
__\( \boldsymbol{A}\cdot \boldsymbol{B}=\frac12(AB+BA) \)__, where __\( AB \)__ is the usual associative product,
and that this
symmetrized product is commutative and satisfies the identity __\( (A^2\cdot B)\cdot
A=A^2\cdot(B\cdot A) \)__, he proposed to consider algebras in which the product
composition satisfies these two conditions. He enlisted the help of
von Neumann and
Wigner in his study, and as a result of their collaboration,
there appeared, in 1934, a paper entitled *On an algebraic generalization
of the quantum mechanical formalism* which was a gem in the structure
theory of algebras. In this paper
[e9],
Jordan, von Neumann and Wigner
obtained a complete determination of the finite dimensional (nonassociative)
algebras over __\( \boldsymbol{R} \)__ satisfying the following conditions: I. Formal
reality in the sense of
Artin–Schreier, that is, the requirement that the only
relations of the form __\( \sum a^2_i=0 \)__ in the algebras are the trivial ones in which
every __\( a_i=0 \)__. II. Commutativity of the product __\( (ab=ba) \)__ and the identity
__\( (a^2b)a=a^2(ba) \)__.

They showed that the algebras satisfying these conditions are direct sums of ideals that are simple algebras, and they determined the simple ones as belonging to one of the following classes:

The vector space over

__\( \boldsymbol{R} \)__of__\( n\times n \)__hermitian matrices with entries in__\( \boldsymbol{R} \)__,__\( \boldsymbol{C} \)__, or Hamilton’s quaternion algebra__\( \boldsymbol{H} \)__, endowed with the algebra structure in which the product is__\( a\cdot b=\frac12(ab+ba) \)__in terms of the usual matrix product__\( ab \)__.The algebras over

__\( \boldsymbol{R} \)__with bases__\( (1,e_1,e_2,\dots ,e_n) \)__and multiplication defined by the table__\( e_ie_j=\delta_{ij}1 \)__, 1 the unit.The algebra

__\( M^8_3 \)__of__\( 3\times 3 \)__hermitian matrices with entries in the algebra__\( \boldsymbol{O} \)__of Cayley numbers, endowed with the product__\( a\cdot b=\frac12(ab+ba) \)__, where, as before,__\( ab \)__is the usual matrix product.

Conversely, the algebras listed satisfy the above conditions.

Now one defines a *Jordan algebra* over a field __\( \mathfrak{F} \)__ of characteristic
__\( \neq 2 \)__ as an algebra in which the product __\( ab \)__ satisfies
__\begin{equation*}
ab=ba,\qquad (a^2b)a=a^2(ba).
\end{equation*}__
Among these are included the *special* Jordan algebras which are isomorphic
to subspaces of an associative algebra closed under the *Jordan product*
__\( a\cdot b=\frac12(ab+ba) \)__ and regarded as algebras relative to this product. Evidently,
the algebras in Jordan, von Neumann, Wigner’s class (1) are special. It is
easy to see, using
Clifford algebras, that this is the case also for algebras
of their class (2). They conjectured that __\( M^8_3 \)__ is not special, and they proposed
the proof of this as a problem to Albert, who showed that this is
indeed the case: __\( M^8_3 \)__ is an *exceptional* (= nonspecial)
Jordan algebra
[13].

After this brief encounter with the Jordan theory, a number of years elapsed before Albert returned to the subject. In a series of three papers appearing in 1946, 1947 and 1950, Albert [29], [30], [34] developed the basic structure theory of finite dimensional Jordan algebras over a field of characteristic not two. Since it is interesting to observe how the subject evolved in Albert’s hands, we shall give a brief indication of the contents of each of these papers.

In the first one he considered Jordan algebras defined concretely as Jordan
algebras of linear transformations of a finite dimensional vector space,
that is, subspaces of __\( \operatorname{End}\mathfrak{B} \)__ closed under the Jordan product __\( A\cdot B \)__. He
proved analogues for these algebras of Lie’s and
Engel’s theorems on Lie
algebras. Assuming the base field is of characteristic 0 (so trace arguments
can be used) he showed that if the algebras contain no nil ideals they are
direct sums of simple algebras. Moreover, he determined the simple
algebras over an algebraically closed field of characteristic 0. This determination
is quite similar to that of Jordan, von Neumann and Wigner’s
of simple formally real Jordan algebras: Class (2) is unchanged, and the
modification required in the definition of class (1) is that __\( \boldsymbol{R} \)__, __\( \boldsymbol{C} \)__ and __\( \boldsymbol{H} \)__ be
replaced by the split composition algebras of dimensions 1, 2 and 4 over
the given base fields. (These are the algebras which occur in
Hurwitz’s problem on quadratic forms permitting composition.) Naturally there is no
class (3) since the algebras under consideration are special by definition.
Actually Albert determined a more general class of so-called reduced
algebras over an arbitrary field of characteristic 0. However, the result he
obtained in the algebraically closed case is adequate to permit the determination
of all the special simple algebras over an arbitrary field using the
method of descent. This was done by
Kalisch
[e18]
and
F. D. Jacobson and
N. Jacobson
[e20].

In his second paper, Albert dealt with abstract finite dimensional
Jordan algebras over any field of characteristic not two. He showed that
nil algebras of this type are nilpotent in the sense that there exists an integer __\( r \)__ such that the product of any __\( r \)__ elements of the algebra in any association
is 0. He proved that if the characteristic is 0 and __\( \mathfrak{U} \)__ has no nil ideals __\( \neq 0 \)__,
then __\( \mathfrak{U} \)__ is a direct sum of simple algebras, and he determined the simple
ones over an algebraically closed field of characteristic 0. Here, one does
have exceptional algebras, and the only simple one over an algebraically
closed field of characteristic 0 is the analogue of __\( M^8_3 \)__ in which the classical
Cayley algebra is replaced by a split Cayley algebra. We shall refer to this
as the *split exceptional simple Jordan algebra*.

In his third paper, Albert extended these results except for a small
gap (which was filled by Jacobson in
[e26])
to the characteristic __\( p\neq2 \)__
case.

A particularly interesting class of Jordan algebras is that of the finite dimensional
exceptional central simple Jordan algebras. If __\( \mathfrak{F} \)__ is the base field
and __\( \bar{\mathfrak{F}} \)__ is its algebraic closure, then __\( \mathfrak{U} \)__ is in this class if and only if the extension
algebra __\( \mathfrak{U}^{\bar{\mathfrak{F}}} \)__ is the split exceptional simple Jordan algebra previously
defined. The theory of these exceptional algebras is intertwined with a
number of other “exceptional” phenomena, notably, exceptional Lie
groups and exceptional geometries (e.g. Cayley projective planes). Albert
made a number of important contributions to this theory, sometimes in
collaboration with others. We have already noted that his first paper on
Jordan algebras established the exceptional character of __\( M^8_3 \)__. In 1959
Albert and
Paige, in a joint paper
[42],
proved a much stronger result:
__\( M^8_3 \)__ is not a homomorphic image of any special Jordan algebra. As a
consequence of this and a result of
Cohn’s
[e24],
one can conclude that
the free Jordan algebra with three generators is not special.

One can distinguish two types of exceptional simple Jordan algebras:
the reduced ones and the division algebras. The first contains idempotents
__\( \neq 0,1 \)__, and in the second the subalgebras generated by single elements and
1 are associative fields. It can be shown that the reduced ones have the form
__\( \mathfrak{H}(\boldsymbol{O}_3,\gamma) \)__ the algebra of __\( 3\times 3 \)__ matrices __\( A \)__ with entries in some (generalized)
Cayley algebra __\( \boldsymbol{O} \)__, which are __\( \gamma \)__-*hermitian* in the sense that
__\( \gamma^{-1}\mkern1mu{}^t\mkern-1mu\bar{A}\gamma=A \)__,
where __\( \gamma \)__ is a 3-rowed diagonal matrix with entries in the base field, and
__\( \bar{A} \)__ is obtained by replacing each Cayley number entry __\( a_{ij} \)__ by its
conjugate __\( \bar{a}_{ij} \)__. The problem of determining conditions for the
isomorphism of two
such algebras was studied by Albert and Jacobson
[40].
It was shown in
this paper that isomorphism of the Jordan algebras implied isomorphism
of the Cayley algebras occurring in their definitions, and they obtained
some rather complicated supplementary conditions for isomorphism.
These sufficed to give a complete classification of reduced exceptional
simple Jordan algebras over number fields. (See also
[e29],
[e35].)

The first construction of exceptional Jordan division algebras is due
to Albert
[41],
[49].
He showed also that no such algebras exist over
number fields. On the other hand, if __\( \mathfrak{F} \)__ is any field over which there exist
central associative division algebras of degree three (e.g. a number field),
then there exist exceptional Jordan division algebras over the field __\( \mathfrak{F}(t) \)__
obtained by adjoining an indeterminate __\( t \)__ to __\( \mathfrak{F} \)__. Albert used a method of
descent in his study of exceptional Jordan division algebras. Subsequently
considerably simpler “rational” constructions were given by
Tits
([e32], p. 412).

From the abstract point of view, a very natural class of algebras (or
rings) is the class satisfying the power associativity condition: subalgebras
(or subrings) generated by single elements are associative. This includes
Jordan algebras, alternative algebras (defined by the identities
__\( a^2b=a(ab) \)__ and __\( ba^2=(ba)a \)__), associative algebras, and a number of other
interesting types of algebras.

Albert initiated the study of power associative rings (without finiteness
conditions) in a paper
published in
[32].
The conditions of power
associativity are that for any __\( a \)__ one has the power formula
__\begin{equation*}
a^ma^n=a^{m+n},\qquad m,n=1,2,3,\dots,
\end{equation*}__
where __\( a^m \)__ is defined inductively by __\( a^1 = a \)__, __\( a^k=a^{k-1}a \)__. In
[32]
Albert showed
that if the additive group of a nonassociative ring __\( \mathfrak{U} \)__ has no torsion, then
__\( \mathfrak{U} \)__ is power associative if and only if it satisfies the two identities __\( aa^2=a^3 \)__
and __\( (a^2)^2=a^4 \)__. These results were obtained by some clever inductive arguments
based on linearizations of the assumed identities. These linearizations
and commutativity yield also the crucial result that in a commutative
power associative ring the map __\( e_R: x\to xe \)__, determined by an idempotent __\( e \)__, satisfies the quadratic equation __\( (2e_R-1)(e_R-1)e_R=0 \)__. If one
assumes that the additive group admits the operator __\( \frac12 \)__, then one obtains
the
Peirce decomposition relative to
__\begin{equation*}
e:\mathfrak{U}=\mathfrak{U}_0(e)\oplus\mathfrak{U}_{1/2}(e)\oplus\mathfrak{U}_1(e)\quad\text{where}
\ \mathfrak{U}_i(e)=\{x_i\mid x_i e=ix_i\}.
\end{equation*}__
One also has extensions of this to Peirce decompositions relative to orthogonal
idempotents. In Albert’s hands, these Peirce decompositions became
powerful tools for investigating power associative rings. He obtained a
number of striking results by this method. We mention two:

Let __\( \mathfrak{U} \)__ be a simple commutative power associative ring whose additive
group contains no elements of orders 2, 3 or 5 and admits the operator __\( \frac12 \)__. Suppose __\( \mathfrak{U} \)__ contains two nonzero orthogonal idempotents __\( e \)__ and __\( f \)__
such that __\( e+f \)__ is not a unit. Then __\( \mathfrak{U} \)__ satisfies the Jordan identity __\( (a^2b)a=
a^2(ba) \)__
[34].

Any simple alternative ring containing an idempotent __\( e\neq 0, 1 \)__ is either
associative or a Cayley algebra over its center
[36].

The ultimate result on simple alternative rings is due to Kleinfeld [e23], [e27]. This states that all simple alternative rings are either associative or Cayley algebras. Albert’s theorem was used as a step in the first proof of Kleinfeld’s theorem.

In
[37]
and
[41]
Albert proved a generalization for power associative
rings of Wedderburn’s celebrated theorem on the commutativity of
finite associative division rings. Call an algebra over a field *strictly power
associative* if all the algebras obtained by extending the base field are
power associative. Also one defines a (nonassociative) division ring by
the property that the left and right multiplications __\( x\to ax \)__ and __\( x\to xa \)__ are
bijective for any __\( a\neq0 \)__ in the ring. Then Albert proved that any finite
strictly power associative division algebra of characteristic __\( \neq2 \)__ is associative
and commutative. Albert based his proof on the determination of the
simple Jordan algebras over an arbitrary field due to F. D. Jacobson and
N. Jacobson
[e20]
and a result of his own on exceptional Jordan algebras
[41].
(Later
McCrimmon gave alternative proofs which are independent
of the structure theory
[e31],
[e34].)
A number of constructions of nonassociative
and noncommutative division rings are due to Albert
[46],
[37],
[45].
These yield examples of nondesarguesian projective planes
including some finite ones.

Albert had a hand in the discovery of several new classes of simple
Lie algebras of prime characteristic (see
[e25],
[38] and
[e28]).
Recently
these results have taken on added luster because of the discovery by
Kostrikin and
Shafarevitch
[e33]
that these Lie algebras can be regarded as
characteristic __\( p \)__ versions of infinite dimensional Lie algebras which had
occurred in
Élie Cartan’s work on contact transformations.

Albert and his students and followers also studied a number of other classes of nonassociative algebras defined by identities. Until now the results which have been obtained on these appear to be of interest only to specialists in the field. We shall therefore refrain from giving any indication of these results. Albert wrote several papers on general nonassociative theory. In one of these [26] he gave a definition of a radical for any finite dimensional nonassociative algebra. Since the theory of the radical is quite interesting and deserves to be better known than it is at present, we take this opportunity to sketch what we believe is an improved version of this theory.

Let __\( \mathfrak{U} \)__ be a finite dimensional nonassociative algebra over a
field. Then __\( \mathfrak{U} \)__ is called *simple* if __\( \mathfrak{U}^2\neq0 \)__
and __\( \mathfrak{U} \)__ has no ideals __\( \neq0 \)__, __\( \mathfrak{U} \)__. __\( \mathfrak{U} \)__ is
*semisimple* if it is a direct sum of ideals which are simple
algebras. Following the pattern of associative ring theory, it is
natural to define the *radical* __\( \operatorname{rad}\mathfrak{U} \)__ to be the
intersection of the set of ideals __\( \mathfrak{B} \)__ of __\( \mathfrak{U} \)__ such that
__\( \mathfrak{U}/\mathfrak{B} \)__ is simple. This definition implies that if no
__\( \mathfrak{B} \)__’s, such that __\( \mathfrak{U}/\mathfrak{B} \)__ is simple,
exist then __\( \mathfrak{U}=\operatorname{rad}\mathfrak{U} \)__. In any case
__\( \mathfrak{U}/\operatorname{rad}\mathfrak{U} \)__ is semisimple or 0, and
__\( \operatorname{rad}\mathfrak{U} \)__ is contained in every ideal __\( \mathfrak{D} \)__ of
__\( \mathfrak{U} \)__ such that __\( \mathfrak{U}/\mathfrak{D} \)__ is semisimple (see, for
example, Jacobson, *Structure of Rings*, p. 41). This implies
that __\( A\neq0 \)__ is semisimple if and only if __\( \operatorname{rad}\mathfrak{U}=0 \)__.

One obtains important information on an algebra __\( \mathfrak{U} \)__ in looking at its
*multiplication algebra* __\( M(\mathfrak{U}) \)__. This is the subalgebra of the associative algebra
End __\( \mathfrak{U} \)__ of linear transformations in __\( \mathfrak{U} \)__ generated by 1 and the left and
right multiplications (__\( a_L: x\to ax \)__, __\( a_R: x\to xa \)__) of __\( \mathfrak{U} \)__. The centralizer of
__\( M(\mathfrak{U}) \)__ in End __\( \mathfrak{U} \)__ is called the *centroid* __\( C(\mathfrak{U}) \)__ of __\( \mathfrak{U} \)__. The study of __\( M(\mathfrak{U}) \)__ and
__\( C(\mathfrak{U}) \)__ was initiated by Jacobson
[e14]
(see also Jacobson, *Lie Algebras*,
pp. 290–295). Albert’s results on __\( \operatorname{rad}\mathfrak{U} \)__, as we shall show, amount to a
formula for __\( \operatorname{rad}\mathfrak{U} \)__ in terms of __\( \operatorname{rad} M(\mathfrak{U}) \)__. We shall call
__\( \mathfrak{U} \)__ *reductive* if __\( \mathfrak{U} \)__
is a direct sum of ideals which are simple algebras and the ideal
__\( \mathfrak{Z} =\{z\mid\mathfrak{U} z=0=z\mathfrak{U}\} \)__. The elements of __\( \mathfrak{Z} \)__ are called
*absolute zero divisors*.
One can show that __\( \mathfrak{U}\neq0 \)__ is reductive if and only if __\( M(\mathfrak{U}) \)__ is semisimple.
Hence __\( \mathfrak{U} \)__ is semisimple if and only if __\( M(\mathfrak{U}) \)__ is semisimple and 0 is the
only absolute zero divisor in __\( \mathfrak{U} \)__. Now let __\( \mathfrak{N} \)__ be the radical of __\( M(\mathfrak{U}) \)__ and
__\( \mathfrak{R} \)__ the ideal in __\( \mathfrak{U} \)__ such that __\( \mathfrak{R}/\mathfrak{N}\mathfrak{U} \)__ is the ideal of absolute
zero divisors of __\( \mathfrak{U}/\mathfrak{N}\mathfrak{U} \)__. Then __\( \mathfrak{R}=\operatorname{rad}\mathfrak{U} \)__.
Albert’s order of ideas in his paper on the
radical is the reverse of what we have indicated; namely, he uses the
ideal __\( \mathfrak{R} \)__ as his definition of the radical, then proves it has the two basic
properties that __\( \mathfrak{U}/\mathfrak{R} \)__ is semisimple and __\( \mathfrak{R} \)__ is contained in every ideal
__\( \mathfrak{D} \)__ such that __\( \mathfrak{U}/\mathfrak{D} \)__ is semisimple.

For certain important classes of algebras (e.g. associative, alternative,
Jordan), __\( \operatorname{rad}\mathfrak{U} \)__ coincides with the maximal nil ideal. For Lie algebras of
characteristic 0, __\( \operatorname{rad}\mathfrak{U} \)__ is the maximal solvable ideal. On the other hand,
Albert has given an example of an algebra in which __\( \operatorname{rad}\mathfrak{U} \)__ is an associative
field. We could not resist recording here a result on the radical which we
have known for some time. This is a generalization of a well-known
theorem of
Hochschild’s
[e17]
on derivations of associative and Lie
algebras.

If __\( \mathfrak{U} \)__ is a finite dimensional nonassociative algebra over a
field of characteristic 0 then any derivation of
__\( \mathfrak{U} \)__ stabilizes __\( \operatorname{rad}\mathfrak{U} \)__.

This can be proved by using the fact that the Lie algebra __\( \operatorname{Der}\mathfrak{U} \)__ of
derivations is the Lie algebra of algebraic groups of automorphisms of __\( \mathfrak{U} \)__.
A more direct proof, which is applicable also in some situations in characteristic
__\( p\neq0 \)__, can be based on Albert’s definition of __\( \operatorname{rad}\mathfrak{U} \)__. We observe
first that if __\( D \)__ is a derivation in __\( \mathfrak{U} \)__, then
__\begin{equation*}
[D,a_L]\equiv Da_L-a_LD=(Da)_L\quad\text{and}\quad[D,a_R]=(Da)_R.
\end{equation*}__
Hence __\( m\to[D, m] \)__ is a derivation in __\( M(\mathfrak{U}) \)__ which we denote as __\( \widetilde{D} \)__.
If __\( m\in M(\mathfrak{U}) \)__ and __\( a\in \mathfrak{U} \)__, then __\( D(ma) = (\widetilde{D}m)a+m(Da) \)__.
It is easily seen that __\( D \)__
stabilizes __\( \operatorname{rad}\mathfrak{U} \)__ if __\( \widetilde{D} \)__ stabilizes __\( \operatorname{rad} M(\mathfrak{U}) \)__.
Our theorem then follows from
Hochschild’s theorem on associative algebras.

In a paper
[25]
which appeared in 1942, Albert introduced a concept
of isotopy for nonassociative algebras. Let __\( \mathfrak{U} \)__ and __\( \mathfrak{B} \)__ be nonassociative
algebras. Then __\( \mathfrak{U} \)__ and __\( \mathfrak{B} \)__ are called *isotopes* if there exist
bijective linear
maps __\( P \)__ and __\( Q \)__ from __\( \mathfrak{B} \)__ to __\( \mathfrak{U} \)__ and a bijective linear map __\( C \)__ from __\( \mathfrak{U} \)__
to __\( \mathfrak{B} \)__ such that for __\( x, y \in\mathfrak{B} \)__ we have
__\begin{equation*}
xy=C((Px)(Qy)).
\end{equation*}__
If __\( P=Q \)__ and __\( C=P^{-1} \)__, we have __\( P(xy) = (Px)(Py) \)__, so __\( P \)__ is an isomorphism.
Isotopy is an equivalence relation. If __\( \mathfrak{B} \)__ and __\( \mathfrak{U} \)__ are identical as sets, and
__\( C= 1 \)__, we call __\( \mathfrak{B} \)__ a *principal isotope* of __\( \mathfrak{U} \)__.
Define a new multiplication on __\( \mathfrak{U} \)__
by __\( u\circ v=(PCu)(QCv) \)__. This, along with the given vector space structure,
gives a new algebra which is a principal isotope of __\( \mathfrak{U} \)__, and since
__\begin{equation*}
xy = C((Px)(Qy))=C((PCC^{-1}x)(QCC^{-1}y)) = C(C^{-1}x\circ C^{-1}y),
\end{equation*}__
__\( \mathfrak{B} \)__ is isomorphic to the principal isotope defined by __\( \circ \)__. This reduces the
consideration to that of principal isotopes. Albert defined isotopy also for
quasigroups
[27],
and he proved a number of interesting results on isotopy
of algebras and of quasigroups. While these have not played an important
role in structure theory, the concept of isotopy has some importance
in nondesarguesian geometry (see
[44]).

#### IV. Miscellaneous

Albert wrote a number of papers
[6],
[11],
[9],
[8],
[16]
on the structure of field extensions. He was particularly
interested in explicit constructions of cyclic field extensions since these
played an important role in his investigations of the structure of division
algebras. Albert’s results on cyclic extensions are presented in a connected
fashion in Chapter IX of his algebra text *Modern Higher Algebra*. There are
numerous references to these results in *Structure of Algebras*. For the
case of degree __\( p^e \)__ and characteristic __\( p \)__, one has an alternative method due
to
Witt, based on Witt vectors, which provides a better survey of cyclic
and abelian extensions (see for example, Jacobson’s *Lectures in Abstract
Algebra*, vol. III, pp. 124–140). On the other hand, Albert’s results on cyclic
fields of degree __\( p^e \)__ and characteristic __\( \neq p \)__ seem not to have been improved
upon until now.

Albert was fascinated by the problem of minimum number of generators for algebraic structures. He proved [28] that any separable associative algebra is generated by two elements and, with John Thompson [43], proved that the projective unimodular group over a finite field is generated by two elements, one of which has order two.

In a joint paper with
Muckenhoupt
[39],
he proved that for any field __\( \mathfrak{F} \)__,
any matrix of trace 0 in __\( M_n(\mathfrak{F}) \)__ is an additive commutator __\( [A, B]=AB-BA \)__.
This supplemented an earlier result by
Shoda
[e15]
for fields of
characteristic 0.

In
[24]
Albert proved that a finite dimensional ordered division algebra
is necessarily commutative. This does not hold for infinite dimensional
algebras, for Hilbert has given an example in the second edition of his
*Grundlagen der Geometrie* of a “twisted” power series division ring which
is not commutative and which can be ordered. It is interesting to note
that Hilbert’s first attempt to give such an example in the first edition of
*Grundlagen* can be seen to be wrong by invoking Albert’s theorem!

Another pretty result of Albert’s gives a determination of the finite
dimensional absolute valued algebras over __\( \boldsymbol{R} \)__. By this we mean a (nonassociative)
algebra over __\( \boldsymbol{R} \)__ which has a map __\( a\to|a| \)__ into __\( \boldsymbol{R} \)__ with the usual
properties:

__\( |a|\geqq 0 \)__and__\( |a|=0 \)__if and only if__\( a=0 \)__;__\( |a+b|\leqq |a|+|b| \)__;__\( |\alpha a| = |\alpha| |a| \)__for__\( \alpha\in \boldsymbol{R} \)__;__\( |ab|=|a||b| \)__.

It had been conjectured by
Kaplansky that if such an algebra has a unit,
then it is alternative, and hence, by a classical result, it is necessarily
either __\( \boldsymbol{R} \)__, __\( \boldsymbol{C} \)__, Hamilton’s quaternion algebra __\( \boldsymbol{H} \)__, or Cayley’s octonions __\( \boldsymbol{O} \)__.
Moreover, in all cases __\( |a| = |a\bar{a}|^{1/2} \)__, where __\( \bar{a} \)__ is the usual
conjugate. Albert
proved this
[31]
and also showed that if all the conditions except the existence
of a unit hold, then the algebra is an isotope of __\( \boldsymbol{R} \)__, __\( \boldsymbol{C} \)__, __\( \boldsymbol{H} \)__ or __\( \boldsymbol{O} \)__.
This result was extended
[33]
to algebraic algebras over __\( \boldsymbol{R} \)__ not assumed
to be finite dimensional.

Albert’s last published paper
[53] — published posthumously — proves
an interesting theorem on quaternion algebras: If __\( \mathfrak{U}_1 \)__ and __\( \mathfrak{U}_2 \)__ are two
(generalized) quaternion division algebras over a field __\( \mathfrak{F} \)__ and __\( \mathfrak{U}_1\otimes_\mathfrak{F}\mathfrak{U}_2 \)__
is not a division algebra, then __\( \mathfrak{U}_1 \)__ and __\( \mathfrak{U}_2 \)__ have a common quadratic
subfield.

Our recital of Albert’s major achievements gives no indication of his methods or, more broadly speaking, of his mathematical style, which was highly individualistic. Perhaps its most characteristic qualities were the directness of his approach to a problem and his power and stamina to stick with it until he achieved a complete solution. He had a fantastic insight into what might be accomplished by intricate and subtle calculations of a highly original character. At times he could have obtained simpler proofs by using more sophisticated tools (e.g. representation theory), and one can almost always improve upon his arguments. However, this is of secondary importance compared to the first breakthrough which establishes a definitive result. It was in this that Albert really excelled. He regarded himself as a “pure” algebraist and in a sense he was. However, his best work — the solution of the problem of multiplication algebras of Riemann matrices — had its origin in another branch of mathematics. Moreover, he could exploit analytic and number theoretic results when he needed them — as he did in this instance.