This book contains the collected works of A. Adrian Albert, a leading algebraist of the twentieth century. Albert made many important contributions to the theory of the Brauer group and central simple algebras, Riemann matrices, nonassociative algebras, and other topics. Part 1 focuses on associative algebras and Riemann matrices, and Part 2 on nonassociative algebras and miscellany. Because much of Albert's work remains of vital interest in contemporary research, this volume will interest mathematicians in a variety of areas.

Readership

A wide range of mathematicians, physicists, and engineers interested in Albert's work.

Table of Contents

• Preface
• Bibliography
• Professional Biography of A(braham) Adrian Albert
• Ph.D. students of A. Adrian Albert
• A. A. Albert by Daniel Zelinsky
• Abraham Adrian Albert, 1905-1972 by Nathan Jacobson
• Abraham Adrian Albert, A biographical memoir by Irving Kaplansky
• A. Adrian Albert by Israel N. Herstein, Collection of articles dedicated to the memory of Abraham Adrian Albert

• On the rank equation of any normal division algebra
• The rank function of any simple algebra
• On the structure of normal division algebras
• Normal division algebras in \(4p^2\) units, \(p\) an odd prime
• The structure of any algebra which is a direct product of rational generalized quaternion division algebras
• The non-existence of pure Riemann matrices with normal multiplication algebras of order sixteen
• A necessary and sufficient condition for the non-equivalence of any two rational generalized quaternion division algebras
• Determination of all normal division algebras in thirty-six units of type \(R_2\)
• A note on an important theorem on normal division algebras
• New results in the theory of normal division algebras
• A construction of all non-commutative rational division algebras of order eight
• The structure of pure Riemann matrices with non-commutative multiplication algebras
• On direct products, cyclic division algebras, and pure Riemann matrices
• On normal division algebras of type \(R\) in thirty-six units
• On the Wedderburn norm condition for cyclic algebras
• A note on cyclic algebras of order sixteen
• On direct products
• Division algebras over an algebraic field
• The structure of matrices with any normal division algebra of multiplications
• On the construction of cyclic algebras with a given exponent
• Algebras of degree \(2^e\) and pure Riemann matrices
• A construction of non-cyclic normal division algebras
• Normal division algebras of degree four over an algebraic field
• On normal simple algebras
• A note on normal division algebras of order sixteen
• (with H. Hasse), A determination of all normal division algebras over an algebraic number field
• A note on the equivalence of algebras of degree two
• On primary normal division algebras of degree eight
• Non-cyclic algebras of degree and exponent four
• Normal division algebras over algebraic number fields not of finite degree
• Cyclic fields of degree eight
• On the construction of Riemann matrices I
• Normal division algebras of degree \(4\) over \(F\) of characteristic \(2\)
• On certain imprimitive fields of degree \(p^2\) over \(P\) of characteristic \(p\)
• Normal division algebras over a modular field
• A solution of the principal problem in the theory of Riemann matrices
• Cyclic fields of degree \(p^n\) over \(F\) of characteristic \(p\)
• On normal Kummer fields over a non-modular field
• The principal matrices of a Riemann matrix
• A note on the Poincaré theorem on impure Riemann matrices
• On the construction of Riemann matrices II
• On cyclic fields
• Involutorial simple algebras and real Riemann matrices
• Normal division algebras of degree \(p^e\) over \(F\) of characteristic \(p\)
• Simple algebras of degree \(p^e\) over a centrum of characteristic \(p\)
• \(p\)-algebras over a field generated by one indeterminate
• Non-cyclic algebras with pure maximal subfields
• On cyclic algebras
• A note on normal division algebras of prime degree
• On ordered algebras
• On \(p\)-adic fields and rational division algebras
• Division algebras over a function field
• Two element generation of a separable algebra
• Absolute valued real algebras
• Absolute-valued algebraic algebras
• Leonard Eugene Dickson (1874-1954)
• On involutorial algebras
• A property of ordered rings
• On involutorial associative division algebras
• A normal form for Riemann matrices
• New results on associative division algebras
• On associative division algebras (Retiring Presidential Address)
• A note on certain cyclic algebras
• Tensor products of quaternion algebras
• Acknowledgments

• A determination of the integers of all cubic fields
• The integers represented by sets of ternary quadratic forms
• On universal sets of positive ternary quadratic forms
• A note on the Dickson theorem on universal ternaries
• On a certain algebra of quantum mechanics
• Integral domains of rational generalized quaternion algebras
• A note on matrices defining total real fields
• Normalized integral bases of algebraic number fields I
• A quadratic form problem in the calculus of variations
• Symmetric and alternate matrices in an arbitrary field, I
• Quadratic null forms over a function field
• A rule for computing the inverse of a matrix
• Quadratic forms permitting composition
• Non-associative algebras. I. Fundamental concepts and isotopy
• Non-associative algebras. II. New simple algebras
• The radical of a non-associative algebra
• An inductive proof of Descartes' rule of signs
• Quasigroups. I
• Algebras derived by non-associative matrix multiplication
• The matrices of factor analysis
• The minimum rank of a correlation matrix
• Quasigroups. II
• Quasiquaternion algebras
• On Jordan algebras of linear transformations
• The Wedderburn principal theorem for Jordan algebras
• A structure theory for Jordan algebras
• On the power-associativity of rings
• Power-associative rings
• On right alternative algebras
• A theory of trace-admissible algebras
• Almost alternative algebras
• A note on the exceptional Jordan algebra
• A theory of power-associative commutative algebras
• New simple power-associative algebras
• Power-associative algebras
• On nonassociative division algebras
• On simple alternative rings
• On commutative power-associative algebras of degree two
• Rational normal matrices satisfying the incidence equation
• The structure of right alternative algebras
• (with M. S. Frank), Simple Lie algebras of characteristic p
• On Hermitian operators over the Cayley algebra
• A property of special Jordan algebras
• On certain trinomial equations in finite fields
• (with B. Muckenhoupt), On matrices of trace zero
• On partially stable algebras
• (with N. Jacobson), On reduced exceptional simple Jordan algebras
• Addendum to the paper on partially stable algebras
• A construction of exceptional Jordan division algebras
• On the orthogonal equivalence of sets of real symmetric matrices
• A solvable exceptional Jordan algebra
• (with L. J. Paige), On a homomorphism property of certain Jordan algebras
• (with John Thompson), Two-element generation of the projective unimodular group
• Finite noncommutative division algebras
• Finite division algebras and finite planes
• On the collineation groups associated with twisted fields
• On the collineation groups of certain non-Desarguesian planes
• Generalized twisted fields
• Isotopy for generalized twisted fields
• On the nuclei of a simple Jordan algebra
• On exceptional Jordan division algebras
• On some properties of biabelian fields
• On certain polynomial systems
• Unpublished Articles: Some mathematical aspects of cryptography
• (with L. J. Paige) Malcev algebras
• Acknowledgments