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Structure of Algebras
A. A. Albert
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Colloquium Publications
1939; 210 pp; softcover
Volume: 24
Reprint/Revision History:
revised 1961; seventh printing 1994; eighth printing 2003
ISBN-10: 0-8218-1024-3
ISBN-13: 978-0-8218-1024-8
List Price: US$54
Member Price: US$43.20
Order Code: COLL/24
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The first three chapters of this work contain an exposition of the Wedderburn structure theorems. Chapter IV contains the theory of the commutator subalgebra of a simple subalgebra of a normal simple algebra, the study of automorphisms of a simple algebra, splitting fields, and the index reduction factor theory.

The fifth chapter contains the foundation of the theory of crossed products and of their special case, cyclic algebras. The theory of exponents is derived there as well as the consequent factorization of normal division algebras into direct factors of prime-power degree.

Chapter VI consists of the study of the abelian group of cyclic systems which is applied in Chapter VII to yield the theory of the structure of direct products of cyclic algebras and the consequent properties of norms in cyclic fields. This chapter is closed with the theory of \(p\)-algebras.

In Chapter VIII an exposition is given of the theory of the representations of algebras. The treatment is somewhat novel in that while the recent expositions have used representation theorems to obtain a number of results on algebras, here the theorems on algebras are themselves used in the derivation of results on representations. The presentation has its inspiration in the author's work on the theory of Riemann matrices and is concluded by the introduction to the generalization (by H. Weyl and the author) of that theory.

The theory of involutorial simple algebras is derived in Chapter X both for algebras over general fields and over the rational field. The results are also applied in the determination of the structure of the multiplication algebras of all generalized Riemann matrices, a result which is seen in Chapter XI to imply a complete solution of the principal problem on Riemann matrices.

Reviews

"In recent years the theory of algebras and hypercomplex numbers has been an active and fertile diversion of modern mathematics. The new book by Professor Albert is therefore a very timely and valuable document. The book gives an extensive account of the present state of the theory including the most recent development. It appears that the whole domain has reached a more mature and clarified form through this work. Anybody familiar with the subject will detect considerable improvement even in some of those parts of the theory which one now considers classical. The value of the book is further enhanced by a complete biography."

-- Mathematical Reviews

Table of Contents

  • Fundamental concepts
  • Ideals and nilpotent algebras
  • The structure theorems of Wedderburn
  • Simple algebras
  • Crossed products and exponents
  • Cyclic semi-fields
  • Cyclic algebras and \(p\)algebras
  • Representations and Riemann matrices
  • Rational division algebras
  • Involutions of algebras
  • Special results
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