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Originally published in 1914, this book provides a concise account regarding the theory of linear associative algebras.
List of contents
Preface; Part I. Definitions, Illustrations and Elementary Theorems: 1-2. Ordinary complex numbers, number fields; 3-4. Matrices; matric algebra as a linear algebra; 5. Definition of linear algebras, coordinates, units; 6-8. Division; principal unit; transformation of units; 9-10. Equations and polynomials in a single number; 11. Unique place of quaternions among algebras; 12-13. Properties of quaternions, relation to matrices; 14. Cayley's generalization of real quaternions; 15-17. Characteristic determinanats; invariance; 18. Binary algebras; 19. Rank and rank equations of an algebra; 20. Ternary algebras with a modulus; 21-22. Reducibility, direct sum, direct product; 23-24. Units normalized relative to a number; example; Part II. Revision of Cartan's General Theory of Complex Linear Associative Algebras with a Modulus: 25-27. Units having a character; example; 28-34. Nilpotent numbers, normalized units; examples; 35. Separation of algebras into two categories; 36-38. Algebras A1 of first category, nilpotent algebras; 39. Normalized units for A1; 40-45. Algebras A2 of the second category; 46. Any A2 has a quaternion sub-algebra; 47-48. Normalized units for A2; determinant; 49. Invariant sub-algebra, simple algebras; 50-51. Main theorem; commutative case; Part III. Relations of Linear Algebras to Other Subjects: 52. Linear associative algebras and linear groups; 53. Linear associative algebras and bilinear forms; 54. Relations of linear algebras and finite groups; 55. Dedekind's view for commutative associative algebras; Part IV. Linear Algebras over a Field F: 56. Statement of main theorem; real simple algebras; 57. Algebras of Weierstrass; 58-60. Division algebras; 61. Statement of further results; 62. Analytic functions of hypercomplex numbers.
Summary
Originally published in 1914 as number sixteen in the Cambridge Tracts in Mathematics and Mathematical Physics series, this book provides a concise account regarding the theory of linear associative algebras. Textual notes are incorporated throughout. This book will be of value to anyone with an interest in algebra and the history of mathematics.
