**Colloquium Publications**

Volume: 24;
1939;
210 pp;
Softcover

MSC: Primary 16;

**Print ISBN: 978-0-8218-1024-8
Product Code: COLL/24**

List Price: $61.00

AMS Member Price: $48.80

MAA Member Price: $54.90

**Electronic ISBN: 978-1-4704-3172-3
Product Code: COLL/24.E**

List Price: $57.00

AMS Member Price: $45.60

MAA Member Price: $51.30

# Structure of Algebras

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*A. A. Albert*

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 & Endorsements

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

# Table of Contents

## Structure of Algebras

- Cover Cover11
- Title page i2
- Dedication iii4
- Preface v6
- Contents ix10
- Fundamental concepts 114
- Ideals and nilpotent algebras 2033
- The structure theorems of Wedderburn 3750
- Simple algebras 4962
- Crossed products and exponents 6578
- Cyclic semi-fields 7891
- Cyclic algebras and 𝑝-algebras 93106
- Representations and Riemann matrices 110123
- Rational division algebras 129142
- Involutions of algebras 151164
- Special results 171184
- Bibliography 193206
- Index 207220
- Back Cover Back Cover1224