1.1. From the Preface.

During the present century modern abstract algebra has become more and more important as a tool for research not only in other branches of mathematics but even in other sciences. Many discoveries in abstract algebra itself have been made during the past ten years and the spirit of algebraic research has definitely tended toward more abstraction and rigour so as to obtain a theory of greatest possible generality. In particular the concepts of group, ring, integral domain, and field have been emphasised.

The notion of an abstract group is fundamental in all science, and it is certainly proper to begin our subject with this concept. Commutative additive groups are made into rings by assuming closure with respect to a second operation having some of the properties of ordinary multiplication. Integral domains and fields are rings restricted in special ways and may be thought of as respective generalisations of ordinary integers and rational numbers.

These fundamental concepts and their more elementary properties are the basis for modern algebra. They are certainly abstract notions but their ultimate absorption by the reader of modern algebra is absolutely necessary and the best place for them is at the beginning. This mode of presentation has not been used in the present textbooks on algebra in the English language but is the customary presentation in all of the more recent texts in foreign languages. We treat the concepts in our first two chapters and present what is basic not only for what follows in the text but for all modern algebra and algebraic number theory.

Our exposition continues in Chapters III, IV, and V with the theory of matrices with elements in a completely general field. Recent trends in algebraic investigation have made it important to know the extent of the validity of the classical theorems on matrices. It is no more difficult to carry out the proofs, where they are valid, for general fields instead of the classical case of subfields of the field of all complex numbers. But it is true that the proofs and results of the classical theory are not always valid. This is brought out clearly in Chapter V, where it is necessary to restrict the types of fields considered.

Essential clarifications in the arguments used in proving matrix theorems are obtained here by extensive use of elementary transformations. These transformations are familiar, except in name, to any reader who has had college training in the theory of determinants and make our proofs of a non-computational character and easy to understand. The author has also attempted to give as much as possible of the algebraic manipulative technique which he has used in his recent investigations in algebras and on Riemann matrices.

The final chapter on matrices presents a rather novel complete generalisation of the theory of symmetric matrices which arises naturally in the algebraic geometric study of Riemann matrices. It is here that even the elementary theorems on symmetric matrices are not valid unless the fundamental assumption is made that the so-called characteristic of the field is not two.

The Galois theory is of great importance in algebra and the theory of algebraic numbers. The older treatments defined the Galois group of an equation with distinct roots as a certain group of permutations on these roots. This treatment is not very simple and the final theorems not in a very good form for certain algebraic applications. One may say that the essential trouble is that the Galois group is defined as a subgroup of the group of all permutations. The more modern treatment is that of the theory of the Galois group of a normal field. This is the set of all automorphisms of the given field and the fact that we take all automorphisms makes our proofs quite simple. We present this treatment in Chapters VI-IX. The first of these chapters gives all of the results needed from the theory of finite groups. Chapter VII gives the essentials of the theory of algebraic extensions of a given field, and in Chapter VIII the Galois theory of fields is given and applied to obtain as a consequence the Galois theory of equations. The final chapter of this set is an application of the theory to obtain structure theorems for the simplest type of algebraic extension, the cyclic field.

The theory of linear associative algebras is a fundamental, if quite advanced, branch of modern algebra. It is natural, however, to introduce this subject from the matrix point of view and we do so in Chapter X. Many quite abstract notions are made concrete by such a treatment, and a quite adequate introduction to the theory is made in this way without going at all deeply into the abstract structure theorems on algebras.

Our exposition closes with an introduction to the theory of $p$-adic numbers. This subject is best studied by considering the general theory of fields with a valuation, and we do so here. The author has collected his material from a large number of sources and hopes that the present exposition is an adequate foundation of the theory. It is certainly true that no progress can be made in reading modern papers on algebraic numbers and their applications to the theory of algebras without a knowledge of $p$-adic number theory. It is equally true that it has heretofore been necessary to read a forbidding number of articles in order to get even a meagre acquaintance with the theory.

1.2. Review by: James Waddell Alexander.
Science Progress (1933-) 33 (131) (1939), 576.

Research in algebra has progressed rapidly on the abstract side in recent years and this book gives an account of the present state of the subject. Prof Albert has himself contributed largely to the theory, especially on the subject of Riemann matrices.

The first two chapters deal with the basic concepts: group, ring, integral domain and field. Then come three chapters on the theory of matrices whose elements belong to a general field.

The Galois theory is taken next. The author develops this from the point of view of the group of all automorphisms of a normal field. This is considered an improvement on the older treatment in which the Galois group arises as only a subgroup of a group instead of as a whole group.

There follows an introduction to the theory of linear associative algebras, an important branch of modern algebra. This is the theory of linear sets $U = (u_{1}, u_{2},..., u_{n})$, of order $n$, defined over a field $F,$ which have been made into associative rings over $F$ by defining multiplication in $U$ suitably. The treatment is from the matrix point of view and this justified by a theorem that every algebra $U$ over $F$ is equivalent to an algebra of square matrices with elements in $F$.

The last two chapters deal with a subject new to textbooks in English, namely the valuation theory of fields and the theory of $p$-adic numbers. This theory involves infinite sequences and what amounts to the notion of a limit; thus it is sometimes called a transcendental theory of fields. The idea of a valuation comes to this: a field $F$ is said to have a valuation ψ if there exists a real function $\psi (x)$, defined over $F$, in the real number field, such that $\psi (0) = 0, \psi (x) > 0$ if $x ≠ 0$ in $F, \psi (xy) = \psi (x) \psi (y), \psi (x + y) ≤ \psi (x) + \psi (y)$. The derived field $F_{\psi}$ of a field $F$ is the least extension of $F$ such that every sequence of elements in the extension, whose values converge in the Cauchy manner, converges to an element of the extension. The valuation $\psi$ is non-Archimedean if and only if $\psi (x + y)$ is at most the maximum of $\psi (x)$ and $\psi(y)$ for every $x$ and $y$ of $F$. The $p$-adic numbers are the elements of the derived fields of algebraic number fields with respect to their non-Archimedean valuations.

The author has a concise style and helps the reader by explaining at suitable stages what is to be done and how much has been done. There are many exercises, designed to illustrate the general concepts in the theory; and at the end is a glossary of the main definitions. The book will be a standard work for those interested in advanced algebra.

1.3. Review by: Archibald Read Richardson.
The Mathematical Gazette 22 (250) (1938), 306-307.

This well printed, moderately priced book by one of the leading algebraists should certainly be acquired by all those interested in algebra. Although, owing to the great progress that has been made during the last ten years, it is not possible in a work of this size to give a complete account of all developments in the subject, some reference, however slight, to important results lying somewhat off the direct line of approach taken by the author might have been justified. Professor Albert's aim has been to build "a foundation for the future exposition of the modern theory of algebraic numbers and class fields and of the theory of linear associative algebra". He achieves his purpose with great clearness and brevity, and the book is of exceptional interest not only as regards its content, but also in its mode of presentation of the subject matter.

Chapter I is concerned with the now customary abstract definitions of groups, rings and fields which render theorems capable of the widest possible applicability.

The main subject of Chapter II is rings having a unity element. Integral domains which generalise the integers and quotient fields which generalise the rational numbers are discussed and the characteristics of rings defined, while in the latter part of the chapter linear algebras are obtained by means of a suitably defined multiplication for linear sets.

Chapters III and IV deal with matrices having elements in a field and with the usual problems arising from the different kinds of equivalence. Properties of the complete matrix algebra consisting of all $n^{2}$-matrices with elements in a field are discussed and the automorphisms of this algebra are shown to be inner automorphisms. Finally the problem of invariance under an automorphism is shown to be equivalent to finding all matrices commutative with a given non-singular matrix.

Chapter V presents a complete generalisation of the theory of symmetric matrices in a field of characteristic ≠ 2. This restriction is necessary, for in a field of characteristic 2 the number $\large\frac{1}{2}\normalsize$ does not exist and -1 = +1.

Chapter VI deals briefly with finite groups and includes the theorems of isomorphism and of Jordan-Hölder. Some results on permutation groups are given at the end of the chapter.

Chapter VII, the subject of which is fields over a basic field $F$, contains the proof of the theorem which today replaces the fundamental theorem of algebra, namely, that a root field exists for a polynomial over $F$. Finite and separable fields are also considered.

Chapter VIII deals with the present conception of the Galois theory of equations, which is based on the theorem that the structure of the group of automorphisms of a normal field, defined as a finite, separable extension of a basic field, in relation to its normal subgroups is simply isomorphic (inversely) with the structure of the field in relation to its normal subfields. At the end of the chapter is a short discussion of solvability by radicals in a non-modular field.

In Chapter IX the different types of cyclic fields are discussed in considerable detail, as they are a subject of great importance not only because they represent the simplest type of field, but also because the theory of algebraic fields since Hilbert has developed principally by the study of class fields for which the Galois group is Abelian.

Chapter X, the subject of which is linear associative algebra as represented by matrix algebras, includes paragraphs on quadratic algebras, cyclic algebras and matrices commutative with a subfield of the ring of all $n^{2}$-matrices.

Chapters XI and XII are two of the most important in the book, as here, for the first time in an English mathematical work, is to be found an adequate introduction to the transcendental theory of fields and of valuation functions, including the theory of $p$-adic numbers. Thus, in this theory the realm of pure algebra is abandoned and limiting processes are defined which will ultimately become of fundamental importance not only for algebra, but also for analysis. The basic idea is that the properties of divisibility in rings are related to topological properties of sets by a generalisation, termed a valuation function, of the concept of the absolute value of a number. Two types of valuation functions are possible, Archimedean and non-Archimedean. Every field with the former type of valuation contains the field of all rational numbers and is equivalent to a subfield of the complex numbers, whereas non-Archimedean valuations lead to the $p$-adic numbers of Hensel...

2.1. From the Preface.

See THIS LINK.

2.2. Review by: Oystein Ore.
Mathematical Reviews MR0000595 (1,99c).

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. The reviewer has only one observation to make which applies not only to this book, but to several others which have recently appeared. It seems preferable, wherever possible, to make the historical development an integral part of the presentation. It adds to the flavour of the book and makes it easier for an unfamiliar reader to decide to whom the various results should be ascribed.

The first chapter gives the more fundamental concepts and lemmas. It is assumed everywhere that the algebras in question have finite bases. One finds the simplest properties of linear sets and transformations, matrix representations, and finally an important lemma by Wedderburn. The second chapter gives the theory of ideals and their principal decomposition theorems. The radical is examined and the existence of idempotents is established preparatory to the proof of Wedderburn's representation theorem. This follows in the next chapter, together with some of the most important properties of normal division algebras, in particular the result that the direct product of a normal division algebra and its reciprocal is a total matrix algebra. Here one also finds the representation of the algebra as a direct sum of the radical and a subalgebra isomorphic to the semisimple difference algebra. Another chapter gives the theory of simple algebras and also the proof that every automorphism is an inner automorphism, together with properties of the maximal subfields and splitting fields. Next follows the theory of cross-products and factor sets giving the construction process for normal division algebras. After a study of the special cyclic semifields one obtains the theory of the important cyclic algebras and $p$-algebras. Very interesting is the chapter on rational division algebras which carries as far as to the proof of the theorem that every normal simple algebra over an algebraic field is cyclic. The remaining chapters contain theories in which the author's work has been of particular importance, namely, the theory of Riemann matrices and the theory of involutory algebras. In a final chapter miscellaneous results on various problems have been collected.

2.3. Review by: Reinhold Baer.
Bull. Amer. Math. Soc. 46 (1940), 587-591.

The study of algebras has been one of the most significant features of the present period in the history of mathematics; and in the theory of algebras a monument has been erected recording some of the characteristic traits of contemporary mathematical thought.

One may say that algebras have been pushed into the centre of attention by the publication of Dickson's Algebras and their Arithmetics; and from that moment on they have kept the interest of the mathematical public. In the meantime a great number of the problems has been solved, methods have been streamlined so that a moment propitious for the survey of the results has arrived. One of the principal actors in the movement has given an account of its results. The mathematical public certainly will be grateful for his effort, as he has been able to capture the inherent beauty of the theory of algebras and to communicate it to the reader.

The book may be divided roughly into two parts, the first being concerned with the general theory, the second containing applications to related problems. It should be mentioned at once that the theory of representations has been "put in its place"; that is, it appears as an application of the general theory and has not been used for the derivation of the results of the general theory.

The general theory of algebras may be defined as that part of the theory in which no special restrictions are imposed upon the field of reference. There are two main topics of discussion. The first is the reduction to simple algebras and the second the discussion of the simple algebras themselves.
...
The most important application of the general theory and perhaps the most interesting part of the whole work is the enumeration of the normal simple algebras over finite algebraic number fields or what amounts to the same thing of the simple algebras over the field of rational numbers. The problem of how to construct all these algebras is settled by the famous theorem that every simple algebra over the field of rational numbers is not only similar to, but may actually be represented as, the crossed product of a cyclic extension of an algebraic number field by its cyclic Galois group.
...
There is a great number of further equally interesting applications of the theory which find treatment in this book. But it would lead too far to discuss them here at great length. Suffice it to mention such topics as the cyclic systems, the modern theory of Riemann matrices and of involutions, and more special problems like the enumeration of all normal division algebras of degrees three and four.

The standard source of reference for the theory of algebras has been in recent years Deuring's report on this topic. As to the material covered none of these works has been a subset of the other and even in their treatment of the common parts they differ widely. In the reviewer's opinion Albert's text is the more easily accessible one of the two, so much so that he feels that the book will serve admirably as an introduction into the theory to all those who know the rudiments of present-day algebra. The expert on the other hand will find this exposition interesting, stimulating and useful both because of the new material included and because of the more "streamlined" treatment of it.

An extensive bibliography of recent publications on related topics has been added which will be welcome and helpful to every student of the field.

2.4. Review by: A R Richardson.
The Mathematical Gazette 24 (258) (1940), 67-68.

Professor Albert is well known for his research in non-commutative algebra and, in particular, in the structure of such algebras, that is, in the relations existing between an algebra and its subalgebras, a subject to which an increasing amount of attention has been given since the appearance in 1926 of the German edition of L E Dickson's Algebras and their Arithmetics. His book contains the first complete and connected account of the subject and also a statement of older theories in modern and simplified form. Whereas in these older theories the algebras are taken over the rational, real or complex numbers, in the modern theory the basic field may be algebraic, $p$-adic or even infinite and quite general. It follows that as a preliminary to the book the reader must be familiar with the ideas underlying the latter as set out clearly in the author's work, Modern Higher Algebra, to which he frequently refers.

The book deals primarily with one of the outstanding achievements of recent years, i.e. the determination of all rational division algebras, that is, of algebras taken over an algebraic number field and having no divisors of zero, and the work of R Brauer, H Hasse, E Noether and the author on the subject. The problem is partly arithmetic in character and the solution depends on properties of the $p$-adic number fields and on the valuation theory, a knowledge of which is now indispensable to algebraists. The excellent account given of these matters in his earlier work is here supplemented by additional results on inseparable fields, splitting fields, $p$-adic algebras and valuation theory, and a clear, simple and connected account of these rather inaccessible subjects and of the solution of the fundamental problem is now available to the student.

Amongst other notable features of the book is the very good explanation which is given of the theory of crossed products, the importance of which lies in the fact that not only are they the only normal simple algebras which have been actually constructed, but also that every normal simple algebra is equivalent to a crossed product. Indeed it is not yet known whether any normal division algebra exists which is not a crossed product.

Professor Albert treats cyclic algebras as instances of crossed products from which the theory of exponents is deduced as well as the consequent factorisation into direct factors of prime-power degree, thus reversing the usual procedure.

Another subject which the author deals with in an original way is the theory of the representations of algebras by matrix algebras over a field. He applies the results to the theory of Riemannian matrices and their generalisations, the study of which leads on to involutorial algebras to which a whole chapter is devoted.

Finally, mention must be made of that part of the book which is really an essay in itself on non-commutative algebra and which includes a very comprehensive and most useful bibliography of 468 works which have appeared in recent years, many of which deal with non-associative algebra now becoming important in physics.

The book is excellently produced and will be indispensable to all workers in this field and to others who want an authoritative account of the theory of linear associative algebras.

2.5. Review by: Morgan Ward.
Science, New Series 95 (2467) (1942), 386-387.

This book, written primarily for specialists in algebra by one of the leading American experts, gives an authoritative account of linear associative algebras which have been the centre of interest in algebra for over sixty years. The book is the first in English to utilise fully the new methods introduced by Emmy Noether and her pupils to refine and extend the theory. The preliminary knowledge necessary for its understanding may be found in the Survey reviewed above, or in the author's own text, "Modern Higher Algebra."

The book begins by giving in less than fifty pages all the classical structure theorems. The remaining three quarters of the book are devoted to the numerous new results obtained in the last fifteen years due in the main to Emmy Noether, Richard Brauer, Hasse and Albert himself. Particularly noteworthy are the

chapters on the representation theory expounding the methods Albert developed in the theory of Riemann matrices and on the structure of rational division algebras where Albert has been able to avoid the complicated arithmetic of integral sets of an algebra. A final chapter, in which numerous unsolved problems are stated, and an excellent bibliography of the recent literature enhance the value of the book for the student.

The book is written with great clarity and precision and more than fulfils the author's stated purpose in the introduction: to provide "a text on the theory of linear associative algebras. (and) a source book for young algebraists." No mathematician at all interested in algebra can afford to miss it.

2.6. Review by: Mathematics and Astronomy.
Nature 145 (30 March 1940), 498.

Not only the contents, but even the title of this book may puzzle many good mathematicians of the older school. Algebra seemed to be a stereotyped subject which started with substitution, addition, subtraction, multiplication and division, had a great deal about equations, progressions, the binomial, exponential and logarithmic theorems, and then, after some chapters on miscellaneous topics, concluded with determinants and the theory of equations. There were a few pages headed the "Fundamental Laws of Algebra", which were generally felt to be an unnecessary statement of the obvious. At one time invariants came into fashion, and were spoken of as the "Modern Higher Algebra".

Now recently books have appeared having scarcely anything in common with older books except their titles. In Prof Albert's Modern Higher Algebra (Cambridge University Press, 1938) the subjects dealt with are groups, rings, integral domains, fields and matrices. The subject of matrices, so important in quantum mechanics and factor analysis, is here taken as the starting point for the discussion of abstract entities satisfying the same laws as certain matrices. The se entities are known as linear associative algebras. In the book under review a more advanced treatment of these algebras is given, including recent advances due to R Brauer, H Hasse, E Noether, and the author himself.

3.1. From the Preface.

During recent years there has been an ever increasing interest in modern algebra not only of students in mathematics but also of those in physics, chemistry, psychology, economics, and statistics. My Modern Higher Algebra was intended, of course, to serve primarily the first of these groups, and its rather widespread use has assured me of the propriety of both its contents and its abstract mode of presentation. This assurance has been confirmed by its successful use as a text, the sole prerequisite being the subject matter of L E Dickson's First Course in the Theory of Equations. However, I am fully aware of the serious gap in mode of thought between the intuitive treatment of algebraic theory of the First Course and the rigorous abstract treatment of the Modern Higher Algebra, as well as the pedagogical difficulty which is a consequence.

The publication recently of more abstract presentations of the theory of equations gives evidence of attempts to diminish this gap. Another such attempt has resulted in a supposedly less abstract treatise on modern algebra which is about to appear as these pages are being written. However, I have the feeling that neither of these compromises is desirable and that it would be far better to make the transition from the intuitive to the abstract by the addition of a new course in algebra to the undergraduate curriculum in mathematics a curriculum which contains at most two courses in algebra and these only partly algebraic in content.

This book is a text for such a course. In fact, its only prerequisite material is a knowledge of that part of the theory of equations given as a chapter of the ordinary text in college algebra as well as a reasonably complete knowledge of the theory of determinants. Thus, it would actually be possible for a student with adequate mathematical maturity, whose only training in algebra is a course in college algebra, to grasp the contents. I have used the text in manuscript form in a class composed of third- and fourth-year undergraduate and beginning graduate students, and they all seemed to find the material easy to understand. I trust that it will find such use elsewhere and that it will serve also to satisfy the great interest in the theory of matrices which has been shown me repeatedly by students of the social sciences.

3.2. Review by: J L Burchnall.
The Mathematical Gazette 25 (265) (1941), 184.

Professor Albert is "fully aware of the serious gap in mode of thought between the intuitive treatment of algebraic theory... and the rigorous abstract treatment of the Modern Higher Algebra as well as the pedagogical difficulty which is a consequence" and the present work attempts to bridge that gap. The result is a work of great interest and charm which will be read with pleasure by many besides the budding specialists for whom it is primarily intended. Many of the concepts of the higher algebra are not difficult to grasp once we are presented with instances of them, and the concepts indeed have their origin in such instances. Some knowledge of the latter is, if not an essential, at any rate a highly desirable preliminary to higher studies. Here they are illustrated by a critical consideration of polynomials, matrices, linear spaces and polynomials with metric coefficients. The final chapter describes briefly the main objects of study in the author's standard work and introduces formally the group, ring, field, integral domain and ideal. For these conceptions the reader's mind is well prepared, though some rearrangement of the sections on ideals and residue classes might elucidate what seems at first sight a somewhat arbitrary definition. The topics of the earlier chapters are fully, though concisely, treated and well illustrate the power and elegance of the methods employed. For this very reason we have some little doubt whether the book quite fulfils its intended purpose. Its special merits will certainly be more keenly appreciated if the reader already has acquaintance with a more extended treatment of matrices and linear forms. As a summary and commentary, however, the work is wholly admirable and, clearly, important.

3.3. Review by: Cyrus Colton MacDuffee.
Science, New Series 93 (2425) (1941), 596-597.

Doubtless the greatest hurdle which a student of mathematics in the United States has to take is the adjustment between undergraduate and graduate work. The amateur student has been replaced by the professional, drill in mechanical skills has been replaced by rigorous thinking, and the attitude that "the proof of this theorem is too hard for us so we shall assume it without proof" is gone forever.

The traditional courses of the senior and first graduate years are critical in the development of a mathematician. It is at this point that keen interest and ambition should be aroused in the student of ability. If uninspired teaching causes him to become bored, he will transfer his interest to another branch of learning. On the other hand, it is only just to a mediocre student to allow him to find out at this point that he will not make a satisfactory graduate student. A mere drill course will not satisfactorily test his abilities.

Elementary courses are best taught from texts, of which there are enough for every taste. Graduate courses are seldom taught from a single text, for the interests of the professor as well as the subject itself are in constant flux. But how are these transitional courses, the "senior" courses, to be handled? Frequently the instructor is not a specialist in the subject, and more frequently he has not the time to work up an adequate set of notes. A good text seems indicated.

But a satisfactory text for a senior course must be more than a rehash of a dozen earlier books. It must be an introduction to modern mathematics. The author must be acquainted with present-day research and he must be able to present his subject in a modern manner, for otherwise the course will not be an introduction to graduate mathematics. A text which is not modern in content and terminology just won't do.

The book under review successfully meets the necessary conditions set forth above. It is specifically designed to serve as an introduction to the author's "Modern Higher Algebra" (which is distinctly a book for graduate students) but can be used effectively by seniors and graduate students who merely wish to know something about matrices.

The book is in two quite distinct parts. The first 108 pages constitute an introduction to the theory of matrices. The usual topics are considered, rectangular matrices and elementary transformations, equivalence of matrices and of forms, linear spaces and polynomials with matrix coefficients. The method of elementary transformations is used wherever possible, even in the proof that the determinant of a product of two square matrices is equal to the product of their determinants. The treatment of determinants is sketchy, for the student is assumed to know how to handle them, but it is lucid and in the modern manner, and should help in dispersing some of the fog emanating from the traditional treatment of determinants. Bilinear and quadratic forms are briefly treated without reference to projective geometry. Linear spaces rationalise the subject of similarity of matrices. Invariant factors and elementary divisors, without which matrix theory is not matrix theory, are treated in Chapter V.

Even though the author states that the book is written for juniors and seniors, it seems to be on the graduate-student side of the fence. It is a fairly solid and comprehensive treatise on the algebra of matrices with no by-lanes or diversions. There is very little motivation for the material, very little to enlighten a student regarding the significance of quadratic forms or similarity of matrices, for instance. There are many new problems in the book, and these will be welcomed by all who teach the subject, but they are mostly drill problems which help not at all in rationalising the subject to an undergraduate. This means that the text must be administered by an understanding teacher. At various places in the text are to be found summaries of further results not treated in detail. Teachers hold differing opinions regarding this as an effective method of teaching, but it should arouse the curiosity of the better students.

This policy of omitting proofs is carried to an extreme in the last twenty-four pages of the book. This last chapter is entitled "Fundamental Concepts," and one is tempted to believe that the author has designed it to serve as a preliminary chapter to his "Modern Higher Algebra." An abstract group is defined, and then one comes up against the statement but not the proof of the simple theorem that the order of a subgroup of a finite group is a divisor of the order of the group. This policy is continued throughout the chapter. After the definition of ring comes the statement, "We leave to the reader the explicit formulation of the definitions of subring and equivalence of rings. They may be found in the first chapter of the 'Modern Higher Algebra."

This last chapter, then, is an encyclopaedic treatment of groups, rings, abstract fields, integral domains, ideals and residue classes, quadratic fields and their integers and the Gaussian field. It is interesting to a mature reader, and under the administration of an expert algebraist should be a quick road to knowledge. A non-specialist who attempts this chapter with a keen class may be in for a few bad moments.

This book is a distinct contribution to the mounting list of books devoted to modern algebra. It is modern in its viewpoint and correct in execution, and the student who has mastered it is on the graduate-student side of the hurdle, ready to pursue further work in abstract algebra.

3.4. Review by: Franklin E Satterthwaite.
Journal of the American Statistical Association 36 (216) (1941), 576.

The purpose of this book is to fill the gap between the standard college algebra course and the abstract algebraic theories such as are treated in the author's book, Modern Higher Algebra. The book is built around the theory of matrices and the algebraic concepts to which matrix theory can be applied. The need for a textbook such as this in English has become quite apparent in recent years. The mathematical statistician is being asked to develop and analyse statistical designs of increasing complexity since the introduction of the analysis of variance. To carry on such research efficiently and to present the results clearly and effectively requires full use of the most powerful and elegant algebraic methods.

The study is introduced with a chapter on polynomials which covers the fundamental definitions, properties, and operations. Chapter II has a similar function with respect to matrices, determinants, and elementary transformations. Particularly valuable are the modern condensed notations and proofs here introduced. Next questions of equivalence are investigated for many general and special types of matrices and forms when they are subjected to different types of transformations. The statistician will recognise in the chapter on linear spaces an abstract treatment of his problems of independence and dependence. The treatment of polynomials with matrix coefficients gives the practical methods by which we may break up complicated statistics such as quadratic forms into fundamental independent components. The last chapter on fundamental concepts introduces the abstract theories of groups, rings, fields, and ideals such as are treated in Modern Higher Algebra.

The book is well written, the theorems are presented in logical order, the proofs are rigorous but never tedious, the notation is compact and clear, and extensive well chosen exercises and numerical problems are included to drive home each point.

This book should be an excellent text to give both the pure and applied mathematician training in the theory of matrices. Advanced undergraduate and beginning graduate students should find it easy to understand. For those who may not be acquainted with Professor Albert, he is one of the leading algebraists in America, having received in 1939 the Cole Prize for the most outstanding research in algebra during the previous ten years.

3.5. Review by: Richard Brauer.
Mathematical Reviews MR0003590 (2,241a).

There is a gap in mode of thought between the usual intuitive first course in the theory of equations and the rigorous abstract treatment of modern higher algebra. To make an appropriate transition from the intuitive to the abstract, Albert proposes adding a new course in algebra to the undergraduate curriculum in mathematics. The book is a text for such a course. It opens with a chapter on polynomials. Then rectangular matrices are treated; the equivalence of matrices and bilinear forms is studied. The fourth chapter deals with linear spaces and linear equations. It is followed by a chapter on polynomials with matric coefficients, elementary divisors, similarity of matrices. The concluding chapter introduces the reader to the fundamental concepts of modern algebra. Groups, rings and fields are defined; the notions of ideals and residue class rings are discussed. The arithmetic in the ring of Gauss integers is developed, and an example of a quadratic field is given in which not every ideal is a principal ideal. In this short review it is not possible to mention all the places in the book where the author gives a new "turn" to known theories. The exposition is careful and clear, and a large number of exercises illustrate the theory. The book forms a very good introduction to Albert's Modern Higher Algebra.

3.6. Review by: Reinhold Baer.
Bull. Amer. Math. Soc. 47 (1941), 849.

Those who have learned abstract algebra know that its methods are at the same time simpler, easier to understand and more penetrating than the more classical procedures; but those who have taught abstract algebra know that there is one great obstacle in the way of the student desirous of acquainting himself with this new approach to algebra. The novice has first to learn to work with concepts, to "compute" with them as easily as he used to do with numbers and such like; though after having succeeded in this, he finds that everything else is comparatively easy going.

It is the object of this Introduction to Algebraic Theories to help the student in his attempt to overcome the difficulties just indicated and to acquire those habits of abstract thinking which are the indispensable foundation not only of abstract algebra, but of almost all present day mathematics. Since this book is intended to be a book of preparation, it does not require any previous knowledge - apart from algebraic techniques already known to the mathematicians of the Renaissance - though a willingness to work and think are indispensable.

The topics treated in this text are those which one would naturally expect: matrices, their equivalence and similarity, as well as a number of related subjects. This entails some study of polynomials - for technical reasons and of linear spaces, since the latter are indispensable for understanding the real significance of the concepts pertaining to matrices. Indications of the technique of generalisation - so important today and of its usefulness are given throughout. The culmination of the whole work may be seen in the last chapter in which the fundamental concepts of abstract algebra are introduced as the natural outcome of the preceding considerations.

A great number of exercises - both numerical illustrations and mathematical applications - provide the reader with an opportunity to test the acquired skills. The book is written with the clarity of style, the arguments are presented with the elegance and precision which one has learned to expect from its author. Thus this text will prove valuable to student and teacher alike.

4.1. From the Preface.

The mathematical material presented in standard texts on college algebra is a fundamental part of mathematics. The concepts involved are basic for an understanding of all more advanced mathematical topics, and the techniques are used in virtually all subsequent courses in algebra and analysis. Nevertheless, college algebra has been a most abused subject. The time allotted to it is frequently inadequate for a genuinely good treatment, and indeed the entire course is sometimes omitted. This is due partly to a desire to bring students to a study of the calculus as early as possible. It is also due partly to the presentation of college algebra, in all texts thus far published, as a collection of seemingly unrelated topics.

The desire to teach the calculus as early as possible tends to defeat its own ends. The building of a course in the calculus on what must be a weak foundation cannot result in a good student understanding of the subject. There is also no reason why the material of college algebra cannot be cohesively organised. About fifteen years ago the author began to study the possibility of reorganising the material of college algebra so as to present it as a sound and unified whole. The result of that study is the present text.

He began with a formulation of what he felt should be standard minimum requirements for all college texts. These requirements are the following:

1. The material should be presented as a unified and compact body of mathematical theory.
   
   
2. The definitions and theorems should be stated accurately. Proofs of results should be given whenever it is reasonable to expect that the better students will be able to grasp them and provided that their inclusion will add to the understanding of the results. Whenever a proof is omitted for any reason it should be made clear that the proof is omitted. Proofs of special cases of theorems should not be presented as proofs of the theorems.
   
   
3. The important techniques and concepts of the text should be emphasised by the presentation of an adequate number of illustrative examples and exercises. Exercises should be given in sufficient numbers so that there are enough both for additional classroom illustration and for student home assignment. They should be solvable by the use of the techniques of the text.
   
   
4. Where such material exists, the text should be rich in additional material on the same subject for the better student. It seems wasteful to use the trick problem as a device to test the student's native ability as a mathematician. His time can be employed far more profitably in the learning of more advanced topics which are a part of the text subject and which are given in the text itself.

The reader may judge for himself how well the present text lives up to these standards. It would seem that they should be met and that few college texts of today meet them.

College algebra has a basic unity. It should consist of a study of the number systems of elementary mathematics, polynomials and allied functions, algebraic identities, equations, and systems of equations. The unity of the present text is achieved by fitting the standard topics of College Algebra into this pattern. Thus the text begins with three chapters on number systems. The first chapter on Natural Numbers introduces the counting concept and so the fundamental idea of a sequence which appears so frequently in mathematics. As permutations and combinations involve nothing but counting, this topic forms a part of the first chapter.

The second chapter presents the theory of the factorisation of integers, the Euclidean greatest-common-divisor process, and a technique for listing the divisors of an integer. These concepts and techniques are a necessary forerunner of the theory of the integral roots of equations with rational coefficients. They are presupposed in most texts on our subject, and there is no basis in fact for assuming that most students have ever seen them.

The theory of rational, real, and complex numbers makes up the third chapter. It is here that the notion of a logarithm, the summation and product symbols, and the concept of a convergent sequence of real numbers are presented. The material should thus present an improved foundation for the calculus.

Chapter IV consists of an introduction to the purely algebraic theory of polynomials and rational functions. All four of the first chapters are written so as to provide the student not only with new material but with a review of the fundamental techniques of elementary algebra. The material usually labelled Mathematical Induction is presented here for what it really is, a derivation of summation formulas by the use of mathematical induction and algebraic identities.

It is then the first part of Chapter V on Identities and Applications. This chapter includes also the identity known as the binomial theorem, and the theories of progressions, which are simply applications of summation formulas.

Chapter VI presents the general theory of polynomial equations as a special case of the theory of factorisation of polynomials.

Chapter VII on the Real Roots of Real Equations contains some techniques usually left for a special course on the theory of equations. A student understanding of these techniques is worth while and the techniques can easily be taught if no attempt is made to derive them.

Chapter VIII on Vectors in the Plane is intended to be treated as optional material. It is hoped that this chapter will provide a basis for a future unification of college mathematics. The chapter begins with a derivation of the parallelogram law for the addition of vectors and the connection of unit vectors with trigonometric functions. The parallelogram law is then used in a derivation of the formulas for the rotation of axes in plane analytic geometry. These formulas yield the addition laws of trigonometry as well as de Moivre's theorem for complex numbers, and they are used to complete the student's understanding of the theory of radicals which began in Chap III. The connection of these topics shows how closely knit college algebra, trigonometry, and analytic geometry really are, and it is hoped that the chapter may be used to inspire a more compact and improved treatment of trigonometry and of plane and solid analytic geometry.

Chapter IX contains the theory of determinants and linear systems. It presents the inductive definition of a determinant and the elementary transformation technique for its computation. There is no reason why a college student should be limited to determinants of two and three rows. Systems of linear equations are solved best by elimination rather than by determinants, and the chapter does not place emphasis falsely on the determinant method.

The final chapter is a pedagogical experiment. There has long been a demand by mathematical economists and psychologists for an early treatment of matrices and quadratic forms. It is also very desirable that such a treatment precede a course on solid analytic geometry. The theory is an advanced mathematical topic only because of the abstract nature of its proofs. The author's personal experience is that the theorems and techniques can be taught to immature students if no attempt is made to derive the results, and so such a presentation has been written here. It is hoped that this chapter will fulfil the needs of the social scientists as well as provide a foundation for an improved treatment of solid analytic geometry.

4.2. Review by: Paul S Dwyer.
Journal of the American Statistical Association 43 (241) (1948), 136-137.

This book incorporates a drastic revision of the conventional approach and treatment of the subject matter of college algebra and as such deserves the attention of statisticians. Important objectives of the author are:

1. The material should be presented as a unified and compact body of mathematical theory. As worked out by the author, this unity is achieved by the successive study of such general topics as the number systems of elementary mathematics, polynomials and allied functions, algebraic identities, equations and systems of equations. Considerable material not usually presented in a course in college algebra is included in this approach.
   
   
2. The definitions and theorems should be stated accurately. This means, of course, that the author has been quite formal in the presentation of the material. This formality will probably be approved by mathematical statisticians. It may not be so welcome to other statisticians who wish to study the main results with as little of the vocabulary and notation of the mathematician as is possible.
   
   
3. Proofs of results should be given only when it is probable that the better students will understand them and when their inclusion will add to the understanding of the results. The author is very careful to indicate each statement which is lacking in proof. In this connection, the final chapter (Chap. 10) on matrices and quadratic forms should be noted since the author makes no pretence at proofs here. This chapter, frankly experimental, is for the purpose of providing matrix material for those who understand the definitions, notation and statements of theorems. As such it will be of interest to many statisticians.
   
   
4. There should be an adequate number of illustrative examples and exercises. The author has provided these and it seems that he has provided another feature which will be of interest to applied statisticians a greater emphasis upon numerical work than is found in most texts on college algebra.
   
   
5. The text should abound with additional material on the same subject for the better student. The author has accomplished this with the use of additional sections and chapters which are indicated as "full course" and which tend to amplify the minimum treatment. In some of this optional material the author attempts to relate the subject matter of college algebra to other material in college mathematics. Chapter 8, "Vectors in the Plane" is of this sort.

...
The author is to be commended for the way in which he has accomplished his objectives. Each reader must determine for himself whether these are appropriate objectives for him. I think that we as statisticians will agree that the emphasis upon numerical work, the more practical approach to the problem of simultaneous linear equations, and the inclusion of matrix material are all steps in the right direction.

4.3. Review by: Oystein Ore.
The American Mathematical Monthly 54 (3) (1947), 174-175.

In his introduction the author justifies his new textbook in college algebra and one can do no better than to let him present his own case: "College algebra has been a most abused subject. The time allotted to it is frequently inadequate for a genuinely good treatment, and indeed the entire course is sometimes omitted. This is due partly to a desire to bring students to a study of the calculus as early as possible. It is also due partly to the presentation of college algebra, in all texts thus far published, as a collection of seemingly unrelated topics. The desire to teach the calculus as early as possible tends to defeat its own ends. The building of a course in the calculus on what must be a weak foundation cannot result in a good student understanding of the subject. There is also no reason why the material of college algebra cannot be cohesively organised."

There is considerable justification for these views. Many college courses and texts have in the main persisted as collections of trick questions with too little emphasis on their systematic background. An excellent example of this type of presentation could until quite recently be found in the preparations for the first actuarial examinations. In algebra as in other fields of college instruction in mathematics there also exists a considerable amount of inertia which tends to make the choice of content in many courses almost dogmatically fixed. There are, of course, certain basic facts which will be with us forever, and which always must be included in the elementary books. But on the whole the newer trends in research and in the applications seem to be slow in exerting their influence. In mathematics there is no revolution as by quantum mechanics in physics, no atom bombs whose explosion immediately makes their impact felt down to the introductory texts.

The need for a revision of the instruction in elementary algebra, both in regard to content and presentation, fortunately has been recognised in a few of the recent texts and the present book is a significant step in the same direction.
...
To sum up, Albert's new textbook should prove a very valuable introduction to college algebra, both through its systematic outlook and through its inclusion of modern topics. The numerous problems add to its usefulness.

4.4. Review by: D B Scott.
The Mathematical Gazette 31 (293) (1947), 61-62.

An elementary textbook by a distinguished authority is always of great interest. The broad background of the expert provides him with a grasp of his subject and a feeling for its principles, which the mere writer of textbooks, however much easier he may be, will probably lack. The study of algebra in this country has for many years been comparatively neglected, and the understanding of even elementary algebraic ideas is not widespread, so that Professor Albert's book is very timely. Although it is designed as a course text for use under American conditions (it is perhaps pitched a little above the level of the intermediate year in our universities), it is certainly relevant to the problems of teachers here.

According to the author's preface, the book is the outcome of fifteen years of reflection on the "possibility of reorganising the material of college algebra so as to present it as a sound and unified whole." There may be few teachers who would not profit from a careful study of this book, though it is not to be expected that many would wish to adopt it in its present form as a textbook for their classes. The interest of the book lies chiefly in the exposition, although the examples are much less feeble than is usual in American works. However, the emphasis on theory rather than examples so long as it does not become a general practice is not in itself a matter for criticism: it is certainly arguable that the only weakness of Hardy's Pure Mathematics is that the magnificent collection of examples it gives serves only too often to divert the student's attention from the desirability of mastering the text

The limited aim of the work has not been allowed to confine the subject matter too closely; every chapter contains optional sections not required for the basic course. The exposition is, on the whole, perfectly clear. There is no possibility of mistake as to which things the author is attempting to prove and which he assumes, but it is unfortunate that these latter are never supported by any specific references to other works. For example, the method used to introduce real numbers is an attractive one, but it is not carried through in detail, and one would have wished to know where to find this done. The layout of the worked examples is noteworthy and wholly admirable, and the whole book has a pleasing appearance, but it is regrettable that the accuracy of the printing is so much below the standard we have come to expect from these admirable publishers.

The book divides fairly naturally into three parts. Chapters, I, II and III on number systems, IV, VI and VII on polynomials and theory of equations, and VIII, IX and X on vectors, matrices and determinants, and sets of linear simultaneous equations. The remaining Chapter (V) on "Identities and Applications" is the least satisfactory part of the book, though it is not easy to say whether this is the cause or the result of its comparative isolation. The exclusion of partial fractions from this chapter is a little difficult to understand. It may be that the author felt that this subject needed to follow the work on simultaneous equations, but the very casual treatment it receives at the end of Chapter IX arouses the suspicion that for all his fifteen years' contemplation - the subject had slipped his mind until a very late stage in the printing. There is, however, little else to criticise. Chapter VI, on equations, might well have been somewhat expanded, and, in particular, more play could have been made with the symmetric functions of the roots; and in the same chapter some mystery is made of the very useful process of "synthetic division" (technically and linguistically an improvement on "detached coefficients"), which some students may perhaps fail to recognise as a shorthand method of carrying out the division of a polynomial by a linear factor.

4.5. Review by: Robert A Melter.
Canadian Mathematical Bulletin 7 (3) (1964), 483.

In many colleges and universities, the traditional course in College Algebra is taught by graduate assistants, if it is taught at all. One of the reasons for the reluctance of senior faculty members to teach the course is the lack of discipline and organisation in the standard textbook. Any mathematician will find it a genuine pleasure to teach from this reprinting of Professor Albert's book.

In the preface the author states: "College Algebra has a basic unity. It should consist of a study of the number systems of elementary mathematics, polynomials and allied functions, algebraic identities, equations, and systems of equations. The unity of the present text is achieved by fitting the standard topics of College Algebra into this pattern." Indeed the standard topics are covered and, one finds, for example, sections in the text on: factorials, permutations and combinations, logarithms, the binomial theorem, arithmetic and geometric progressions, quadratic equations, the remainder and factor theorems, Descartes' rule of signs, and Horner's method.

A feature of the text is the inclusion of topics which could be included in an expansion of the standard course. Among these are: the Euclidean greatest common divisor process for both integers and polynomials, the unique factorization theorem for polynomials, and two chapters on matrices and quadratic forms.

This book can serve as good preparation for a course in modern algebra; the material is presented with care and rigour, and the student is not introduced to ideas which he will have to unlearn at a later date. [The g.c.d. of two integers is defined (p. 40) as their largest positive common divisor. This differs from the definition in Birkhoff and Mac Lane where a g.c.d. of two integers is defined as a common divisor which is a multiple of every other common divisor.] The prospective calculus student will also be well served by the numerous drill exercises, including a substantial number of oral exercises.

5.1. From the Preface.

In a recent text on college algebra the author gave a brief presentation of what seems to him to be the best basis for a modern course on plane analytic geometry, that is, the algebraic vector approach. The present text contains an extension of this approach, yielding an exposition in full for the three-dimensional case, and thereby ties up the study of space analytic geometry with the theory of vectors and matrices.

Chapters 1 and 2 contain a treatment of the equations of lines and planes. After a preliminary study of the linear operations on n-dimensional vectors, rectangular coordinates are introduced, three-dimensional vectors and inner products are interpreted geometrically, and, from a consideration of scalar products and axis translations, the parametric equations of a line are obtained. The vector approach then yields a very simple derivation of the normal form of an equation of a plane, and the standard forms of plane and line equations are rather immediate consequences.

Chapter 3 contains an exposition of classical elementary surface and curve theory. Chapter 4 contains the usual treatment of spheres, and Chapter 5 gives the classical descriptions of quadric surfaces in standard position. The latter chapter ends with a rather novel classification of quadrics according to certain invariants.

Chapter 6 is an exposition of that part of the theory of matrices which is needed for a complete development of the so-called principal axis transformation. A full account of the orthogonal reduction of a real quadratic form in n variables is given, and the theory is applied in Chapter 7 to the three-dimensional case of quadric surfaces. This latter chapter ends with a discussion of the symmetries of quadric surfaces.

The first seven chapters of this text provide an exposition of the basic topics of solid analytic geometry, the material being adequate for a one-quarter course on the subject. The remaining chapters contain additional material for longer courses or outside reading. Chapter 8, on spherical coordinates. contains a discussion of some practical aspects of the theory of rotations and translations of axes in space. It is quite clear that rectangular coordinates are not as practical for actual measurements as are the coordinates of range, azimuth, and elevation, and in this chapter methods are developed for actual computation of the changes in these measurable coordinates after translations or rotations of axes. The chapter ends with a discussion of gnomonic charts.

Our final chapter contains a brief presentation of the elements of projective geometry. The effect on the theory of linear transformations of the use of homogeneous coordinates is given, and the chapter contains a rigorous matrix proof of the invariance of the cross ratio under projective transformations. It is hoped that the use of modern algebraic techniques in this and in the earlier chapters of the present text will serve to make the subject of solid analytic geometry fit better in the teaching of modern mathematics than it has in the past.

5.2. Review by: D B Scott.
The Mathematical Gazette 34 (308) (1950), 143-144.

The successful writer of elementary textbooks is not usually an outstanding expert in the subject on which he writes. It is true that it is useful to know as much, or more, about the subject as is intended to be learnt from the book, but this is not an absolutely indispensable qualification. The real expert does not often succeed in writing a first-rate elementary text, and his books are, possibly because he sees further into the subject, usually much heavier than those of the professional textbook writer. For instance, Sir Horace Lamb's classical texts on Mechanics are rather tougher than those of his successors, of whom Sir James Jeans (whose text on the subject is too frequently overlooked) is about the only one who can compare with him in scientific stature. When the expert does score a resounding success it is mostly because he has something important to say about the direction which elementary teaching should take. It is this which accounts for the success of Hardy's Pure Mathematics, and, because he unburdened himself of what he needed to say in that book, he was under no compulsion to write another book of the same type. Professor Albert's College Algebra, although not so outstanding as Hardy's work, was also written for a real and urgent purpose and was the outcome of long and arduous cogitation on the questions with which it dealt. Unlike Hardy, Albert has not been able to resist the temptation of trying to repeat the success of that book and this new work is the result.

It would be unjust to Professor Albert to suggest that he is not attempting something useful in this book. His object is to bring about a simplification in the presentation of the subject by allowing himself a little more elaboration in the tools and techniques used. This is a highly commendable aim and is the basis of most advances in the presentation and digestion of mathematical theories, and of progress in most other forms of craftsmanship. Nevertheless, it does not appear to the reviewer that the attempt has clearly succeeded. The algebraic techniques used are not unfairly advanced and quite a lot of the exposition is worthy of careful study for its demonstration of the possibilities of the technique of transformation of coordinates. But the treatment is so heavy that the advantages are outweighed by the loss of that crispness and geometrical feeling which should animate an elementary introduction to three-dimensional analytic geometry.

6.1. From the Preface.

The existence of the digital computer and other devices using binary digits has resulted in a renewal of interest in the mathematical theory of finite fields. The main exposition of the foundations of the subject was written over fifty years ago and has been out of print for many years. A new exposition using the modern theory of algebraic extensions of arbitrary fields seems very timely and is the principal reason for the preparation of the present text.

We give here a compact and self-contained exposition of the fundamental concepts of modern algebra which are needed for a clear understanding of the place of finite field theory in modern mathematics.

In the first chapter we introduce the primitive concepts on which algebra is based and go on to a fairly complete exposition of the basic notions about finite groups, including the theory of composition series in general form. In Chapter II we present the concepts of ring, ideal, difference ring, polynomials over a ring, and the simple cases of the integral domain of ordinary integers and of polynomials in one indeterminate over a field. The material of this chapter is then the foundation for the later theory of algebraic extensions.

A logical exposition of the theory of field extension is impossible without a preliminary exposition of the theory of vector spaces, linear mappings, and matrices. This is then the subject of Chapter III. All of the theory of matrices which seems to be appropriate is presented, including the theory of matrix equivalence over a polynomial ring and the theory of similarity. Since the theory of quadratic forms and orthogonal equivalence seems out of context, it is not presented.

In Chapter IV we provide a substantial exposition of the theory of algebraic extensions of fields, including the Artin version of the Galois theory. The chapter ends with a new and remarkably simple proof of the normal basis theorem for cyclic fields. The real reason for the book is Chapter V, in which we present a modern and improved exposition of the foundations of the theory of finite fields. Galois theory and group theory are used where they belong, and many proofs are improved thereby.

It is hoped that this new text will prove to be a valuable aid not only to those who wish to learn the basic theorems on finite fields but to those who wish to refresh their knowledge of modern algebra or to review a series of courses on modern algebra which they may have just completed. The text should also be usable as a first course in the subject, provided that it is recognised that the presentation is exceedingly compact and requires slow and careful classroom discussion.

6.2. Review by: Cyrus Colton MacDuffee.
The American Mathematical Monthly 64 (8) (1957), 602.

The classic book of L E Dickson, Linear Groups with an Exposition of the Galois Field Theory (1901) has gained stature with the passing years. The treatment of linear groups was more complete than even the author probably realised, and the book has been, over the years, the major source of information on finite fields. It has inspired considerable research in the last half century, and furthermore, notations and points of view have changed. It is timely, therefore, to have an up-to-date exposition of the subject of finite fields using the slick proofs of modern algebra and embracing the latest results.

As the author states, the real reason for the existence of this book is the fifth and last chapter of 29 pages. The usual basic theorems on Galois fields are quickly developed. A proof is given of Gauss' theorem determining the integers m for which primitive roots modulo m exist. The Galois group of the finite field is determined. The familiar theorems of Serret and Dedekind on irreducible polynomials are given, and the concept of exponent to which an irreducible polynomial belongs is used to extend and clarify this theory. Now come many detailed results on the construction of irreducible polynomials. The book ends with a discussion of Dickson's method for computing primitive irreducible polynomials, and a list without proofs of eighteen known theorems.

In order to make the book essentially self-contained, the author has put in four preliminary chapters totalling 122 pages. These are entitled: I, Groups; II, Rings and Fields; III, Vector Spaces and Matrices; IV, Theory of Algebraic Extensions. In order to cover so much material in so short a space, the author has employed two devices. He has limited the theory to those topics which are necessary for his Chapter V, and he has in places drastically condensed the usual presentation. Thus Chapter III omits quadratic form theory, and all proofs of theorems on determinants.

It is thus evident that the title of the book is too comprehensive. In the reviewer's opinion the book is not well suited for use as a textbook with beginning graduate students. For those who have been over the material once, it would furnish an excellent review and summary of many parts of modern algebra. It is a scholarly piece of work, with some misprints, to be sure (but what book does not have some?), and is the last word on finite fields.

6.3. Review by: Walter Jacobs.
Science, New Series 124 (3235) (1956), 1296.

For the greater part of his new book, A A Albert reworks the same ground treated in the first nine chapters of his Modern Higher Algebra, written nearly 20 years earlier. However, in the present volume his presentation leads up to a discussion of finite fields, whereas the former volume closed with an account of $p$-adic number fields.

The first chapter presents the elementary theory of finite groups; the second discusses rings, fields, and some basic concepts of ideals; and the third covers vector spaces and matrices. Chapter IV is devoted to finite algebraic extensions of a field and to Galois theory in the modern treatment. The fifth and final chapter applies the methods of the previous chapter in a systematic study of the irreducible polynomials over a finite field. A concluding section of Chapter V lists about 20 theorems of L E Dickson on finite fields, without giving proofs.

Although most of the book reviews the fundamentals of modern algebra, there is more here than a simple repetition of old material. The selection and arrangement are expertly done, and new proofs are produced for a number of theorems to improve the unity and logical structure of the presentation.

Unfortunately, the virtues of the book are likely to be appreciated only by the specialist. Albert's style at its softest makes few concessions to the reader, and in this case, as the author notes in his preface, "the presentation is extremely compact, and requires slow and careful classroom discussion," if it is to be used as a textbook for a first course in modern algebra. A rather liberal sprinkling of typographic errors will add to the student's troubles.
...
It is too bad that the difficulties of style will limit the number of readers who might otherwise appreciate the brilliant qualities of this book. Mathematics could use more writers like G H Hardy.

6.4. Review by: R Rado.
Science Progress (1933- ) 45 (179) (1957), 549-550.

In this book the author has set himself the task of presenting a modern account of the theory of finite fields. Such an account will be widely welcomed since an exposition of that theory can only be found either scattered in various books on other parts of algebra, and then rather condensed and far from complete, or else in L E Dixon's "Linear groups and the Galois Theory" (1900) which was a pioneer work in its day, but must now be considered as rather out of step with contemporary algebra. Since its inception by Galois, in 1830, the theory of finite fields has reached a remarkably high state of development and has given rise to deep and important investigations such as those on the congruence zeta function - obviously outside the scope of this book and all those interested in such fields will be grateful to the author.

The book is self-contained. It presupposes no knowledge of abstract algebra although it requires a considerable amount of work from a conscientious reader who wants to fill in the details deliberately left out by the author. Systematical developments will be found of all those concepts which are needed for a full understanding of the position of finite fields within the general framework of algebra.

Chap. I is devoted to group theory leading up to the Jordan Schreier theorem, Chap. II to rings and fields and contains the elementary theory of ideals and polynomials. Chap. III deals with vector spaces and matrices comprising basic facts on linear transformations and canonical forms of matrices. Chap. IV contains a detailed treatment of algebraic extensions of fields, including Artin's development of the Galois Theory and a new proof of the normal basis theorem. With Chap. V the main topic of finite fields is reached and is thoroughly considered. Full use is made of the preceding material. Much attention is paid to concrete constructions of polynomials of prescribed degree which are irreducible over a given finite field. Each section concludes with a number of helpful exercises, and short historical notes add to the interest of the book.

6.5. Review by: R L Goodstein.
The Mathematical Gazette 42 (340) (1958), 154.

This little book presents a remarkably clear and concise account of the theory of algebraic extensions of a field, including the version of Galois theory evolved by Artin, and a new exposition of the foundations of finite field theory making extended use of Galois and group theory. The classical problem of establishing the existence of the roots of unity in a given finite field is obviated by the introduction of the theory of splitting fields; a field $K$ is said to split a polynomial $\phi(x)$ in an indeterminate $x$ over a field $F$ if $\phi$ may be resolved into its linear factors in $K$, any two splitting fields being isomorphic.

The treatment of logical questions in the foundations of algebra is much more satisfactory than in most texts on modern algebra. There is, for instance, none of the usual vagueness about indeterminates.

6.6. Review by: Arthur Mattuck.
Bull. Amer. Math. Soc. 63 (1957), 323-325.

Finite fields are soiled: computing machines are beginning to use them. Dickson's Linear groups and the Galois theory is the classical exposition of the subject, but since it was written modern algebra has come into existence; Albert's avowed purpose is therefore to give us a timely, modern version of the theory, setting it within its proper context as part of general field theory. But books behave waywardly while they are being written; in a sort of Tristram Shandy fashion four-fifths of this one is over before we get around to the finite fields, and then what we do read about turns out to be a rather brief and odd assortment of material - perhaps representing what will be most useful to those with practical applications in mind. On the other hand, the main part of the book consists of a good, compact presentation of the essentials of modern algebra. As such, it has a wide potential audience and has in fact been written with a wide class of readers in mind. In short: a nice, lightweight algebra text with an addendum on finite fields.

A great deal is covered in its 150-odd pages. The first two chapters discuss in turn group theory (no operators) through the Jordan-Hölder theorem and basis theorem for abelian groups, then elementary ring and ideal theory, with a discussion of factorisation for polynomials in one variable. Chapter three treats vector spaces and matrices; elementary transformations of matrices, the characteristic function, and elementary divisor theory are discussed in some detail and there is in the problems a heavy classical emphasis on matricial computations: triangularisation, determination of rank, inverse, and invariant factors. The fourth chapter deals with field extensions and Galois theory, concluding with the normal basis theorem. The detailed study of particular fields is the only way to learn Galois theory; all algebra students should be directed to the more than two pages of specific computational problems about fields of all sorts that Albert includes here.
...
The universal aims of this book place it more or less inevitably in the please-all-of-them-some-of-the-time category: everyone is offered a berth, but no one will find the fit quite exact. Those wanting to learn or review algebra will be happy to find that in remarkably few pages all the essentials are treated, and they will (or ought to be) delighted with the problems, which will help self-study greatly. But in the condensation the leisurely inter- and intra-theorem explanations of, say, the author's Modern higher algebra have had to be sacrificed, and the proofs have become little pieces of old hen: nourishing, but tough and chewy. Albert is well aware of this, of course; he has elected to write his book in a certain style and in so doing he is not without plenty of distinguished predecessors. My own feeling is that it is a poor style for a textbook - especially one for the generally undisciplined American. If the book accompanies a course or is being used for review - his own suggestions for its use then this is perhaps not such a very serious matter, and after all, the theorems do in a sense speak for themselves. I think however that most students welcome toastmasters who are willing to say a little more than the inevitable and perfectly useless cant, "We have the following result... we next prove...."

Those interested in finite fields will find the last chapter interesting and presumably useful; on the other hand they might wish the preceding exposition trimmed of the material which is not really relevant (all the matrix theory, for instance) and to see the extra space used to expose the finite fields in a more complete, more organised, and more explanatory fashion. The computers perhaps have no use for it, but I for one was pained by the omission of Chevalley's theorem that a finite field is quasi-algebraically closed ("Demonstration d'une hypothèse de M Artin," which Albert points out was actually conjectured some 25 years earlier by Dickson - but Artin's reasons were elegant and convincing). It is a theorem which goes significantly into the structure of the finite field, and at the same time it has one of the most beautiful proofs in algebra. It seems that the machines are to crush our daisies, too.

7.1. From the Publisher.

Following a review of the basics of projective geometry, this text for advanced undergraduate and graduate students proceeds to discuss finite planes, field planes, and coordinates in an arbitrary plane. Additional topics include central collineations and the little Desargues' property, the fundamental theorem, some non-Desarguesian planes, and an appendix on the Bruck-Ryser theorem.

7.2. From the Preface.

In this book the authors have endeavoured to introduce the subject of finite projective planes as it has developed during the last twenty years. It is a rich and rewarding area of mathematics, whose study uses a wide variety of techniques - geometric, algebraic, combinatorial, etc. Also, it is an area containing a great many unsolved problems which require no complicated jargon to state, and which are understandable to the student who has just begun to study the subject. Finally, as is the case with much of number theory, a good deal of the study of projective planes can be taught to the student who is at an early stage in his mathematical development, and the subject can be given an elementary and self-contained presentation.

We have attempted to provide as elementary an approach as is possible consistent with the inclusion of all of the theorems we felt belonged in this book. Thus, the notion of a projective plane is introduced by way of the seven point (Fano) plane and the real projective plane, and the incidence properties of these planes are generalised to provide the axioms defining our subject. Then duality, and the axiom of Desargues are discussed along with other elementary geometric concepts. Next, the finiteness assumption is made, and several combinatorial properties of finite planes are examined in some detail, both for their own intrinsic interest and beauty, and for their utility, in leading up to the main results.

As the geometry unfolds, enough of the study of the relevant algebraic systems is covered to make the book truly self-contained. In fact, the only theorems used without being proved in the present volume are the classification theorem for finite fields, and Weddenburn's theorem on division rings. The algebraic topics introduced include loops, groups, fields, matrices, and ternary rings. Thus, the algebra introduced permits us to make a study of the collineation groups of planes, and, in particular, of the projective group of the classical finite planes defined by Galois fields.

The deviation from the classical study of the subject first occurs when the planar ternary ring is introduced as the "coordinate system" of the arbitrary projective plane. From that point on, a careful study is made of the geometric consequences of assuming more and more algebraic structure in the coordinatising planar ternary ring of a plane until we have obtained sufficient information to prove our fundamental theorem: A finite plane can be coordinatised by a Galois field if and only if the plane is Desarguesian. Along the way, there is unfolded for us a set of theorems detailing the relationship between the algebraic conditions in a planar ternary ring and the geometry of the plane it coordinatises. This lays the foundation for a more thorough study, in the mathematical literature, of groups, loops, near fields, Veblen-Weddenburn systems, etc. .., in order to be able to understand and to possibly classify all finite projective planes in the distant future.

The book concludes with some examples of finite non-Desarguesian planes, which can easily be described in the same elementary terms as the earlier exposition.

It is the authors' contention that the subject of projective planes can be taught at virtually any undergraduate or graduate level. At the most advanced levels, of course, there is no question of this. It is a rich and interesting area of mathematics with an extensive research literature. The possibility of presenting as elementary an exposition of the subject as the present one, however, proves it to be admirably suited for teaching to undergraduates with little or no previous exposure to mathematical ideas. In fact, it seems to us to be an excellent vehicle for the introduction of a variety of mathematical concepts and theorems to the mathematical novice. Accordingly, we have attempted to write in such a way as to make the book teachable at an elementary level. We assumed virtually no knowledge or sophistication on the part of the student, and defined and discussed every algebraic system used as it became necessary. The more advanced student could speed up his study of the book by omitting those sections containing an exposition of subjects he was already familiar with. Finally, the working mathematician should be able to find here an exposition of a part of a beautiful and fascinating area of mathematics.

There are many exercises scattered throughout the book. Some require that the student complete the proof of a theorem which has been only partially proved, some ask the student to delve more deeply into the properties of the algebraic systems which are introduced, and many are simply interesting problems concerning projective planes. Taken as a whole, the problems are a significant tool for the understanding of the subject and of the mathematical tools necessary for its study.

7.3. Review by: Frederick Walter Wilke.
Mathematical Reviews MR0227851 (37 #3435).

In this little book the authors have attempted to lay the groundwork for a study of finite projective planes. In the reviewer's opinion they have succeeded quite well. The book is essentially self-contained and can be read easily by anyone who has had a little experience with abstract mathematical systems. Because of the abundance of easily constructed examples, the book could also be used as a vehicle for introducing a student to abstract mathematics.

The reader is led from basic definitions and examples to the construction of the field planes and on to the coordinatisation of an arbitrary projective plane by a planar ternary ring. Connections between the geometric properties of the plane and the algebraic properties of the ternary ring are stressed. The Desargues and little Desargues properties are discussed and connected with the existence of certain central collineations. The foregoing all leads up to the theorem that a finite projective plane is Desarguesian if and only if it is a field plane. The book closes with a short chapter giving some examples of non-Desarguesian planes, and an appendix which gives a brief discussion of the Bruck-Ryser theorem. The exercises form an integral part of the presentation and should be worked diligently by anyone who wants to get a "feel" for the subject.

7.4. Review by: Fletcher Gross.
The American Mathematical Monthly 77 (1) (1970), 92-93.

The study of finite projective planes, besides being of interest for its own sake, provides an excellent introduction to many other areas of mathematics. For example, combinatorial problems, groups, and matrices all arise naturally in a projective geometry context. The authors' intention in the present book is to provide an introduction to the theory of finite projective planes that would be accessible to students with little or no background.

The first two chapters cover basic definitions and combinatorial results. Chapter 3 is concerned with planes constructed over finite fields and introduces matrices as a means of representing collineations. Coordinates in an arbitrary plane and the resulting ternary rings are introduced in chapter 4. After a treatment of central collineations in chapter 5, the high point of the book occurs in chapter 6. The main result proved here is that a finite plane is Desarguesian if, and only if, it can be coordinatised by a field. Chapter 7 provides examples of finite non-Desarguesian planes along with some miscellaneous results on groups and fields (for example, Lagrange's theorem on finite groups is included).

The treatment throughout this book is largely from an algebraic point of view. In the reviewer's opinion, the sort of course for which this book is intended would be a good preparation for the usual upper-division undergraduate course in modern algebra. The book is self-contained with the exception of three results stated without proof: Wedderburn's theorem on finite division rings, the classification of finite fields, and the Bruck-Ryser theorem on the nonexistence of finite planes of certain orders (this last theorem is stated only in an appendix and is never used).

The reviewer's main criticism of the book has to do with certain omissions. Instead of stating without proof that there exist infinite non-commutative division rings, why not simply give the example of the real quaternions? Also the treatment of finite fields seems altogether too skimpy. Instead of just stating a theorem without proof, why not at least give the student some indication of how finite fields of other than prime order can be constructed. In addition, it is unfortunate that the book includes nothing about the relationship between finite planes and other areas of combinatorial theory such as orthogonal latin squares. In short, the intent of the book is a worthy one, but I wish the book were longer and the material a little more developed.

7.5. Review by: W C Ramaley
The American Mathematical Monthly 77 (1) (1970), 93.

Until this book, any discussion on the undergraduate level of the existence of non-Desarguesian planes had to centre on free planes. But since many elementary examples of projective planes are finite, a student is apt to ask about finite projective planes. I used this book as a text at the advanced undergraduate level.

The topics covered include both recent work and traditional results. The necessary algebraic concepts are introduced, but wisely there are no attempts to prove the Wedderburn or Bruck-Ryser theorems. The students may tire of co-ordinatisation theorems at times, but this is the core of the material and the theorems are done as effortlessly as possible. The problems are of a routine type intended more to clarify than to amplify the text.

Used alone, this book is insufficient as a text for most geometry courses, even for most projective geometry courses. Some synthetic and some real projective geometry needs to be done in an undergraduate course. But if this book is used to augment a more standard text, it provides a rich development heretofore impossible on the undergraduate level. At the minimum, everyone concerned in teaching projective geometry and wishing to discuss the finite case should have this book available.

7.6. Review by: A Hedayat.
Biometrics 26 (1) (1970), 162-163.

Consider a system containing a set of distinct elements called 'points' and certain subsets of them called 'lines' together with an incidence relation (a point incident with a line or a line incident with a point). The system of points and lines is said to form a projective plane if the following axioms are satisfied: (1) To any two distinct points, there exists a unique line incident with both of them. (2) To any two distinct lines, there exists a unique point incident with both of them (no parallel lines exist in the system). (3) There exist four points of which no three are incident with the same line.

A projective plane is called finite provided it contains only a finite number of points. The equivalence of finite projective planes with other combinatorial systems such as orthogonal Latin squares, balanced incomplete block designs, partially balanced incomplete block designs, orthogonal arrays, error correcting codes, nets, difference sets, mutually orthogonal matrices, graphs, etc. has made this subject very interesting.

It is not easy to write an adequate book with less than 100 pages about finite projective planes. Albert and Sandler have succeeded in preparing such a book. Their book should be welcomed because it is well motivated and well-written. It is an elementary and self-contained introduction to the subject.

The book contains 7 chapters and an appendix. Chapter 1 introduces and discusses projective planes, subplanes, incidence structure, isomorphism of planes, duality, and the principle of duality, and Desargues' configuration.

Chapter 2 deals with some counting lemmas, the order of finite projective planes, loops and groups, collineations, the incidence matrices and some other combinatorial results. I believe the inclusion of a collineation which is an involution and also a loop, but which is not a group, would have benefitted the reader who for the first time faces such topics.
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