The present book is meant as a basic text for a one year course in algebra, at the graduate level.
Unfortunately, the amount of algebra which one should ideally absorb during this first year in order to have a proper background (irrespective of the subject in which one eventually specializes), exceeds the amount which can be covered physically by a lecturer during a one year course. Hence more material must be included than can actually be handled in class.
Many shortcuts can be taken in the presentation of the topics, which admits many variations. For instance, one can proceed into field theory and Galois theory immediately after giving the basic definitions for groups, rings, fields, polynomials in one variable, and vector spaces. Since the Galois theory gives very quickly an impression of depth, this is very satisfactory in many respects.
One can also treat first the linear algebra, after covering the basic definitions, and postpone the field theory. The chapters have been so written as to allow maximal flexibility in this respect, and I have frequently committed the crime of lèse-Bourbaki by repeating short arguments or definitions to make certain sections or chapters logically independent of each other.
I have followed Artin in the treatment of Galois theory, except for minor modifications. The reader can profitably consult Artin's short book on the subject to see the differences. Furthermore, the reader would also profit from seeing an exposition based on the Jacobson-Bourbaki theorem, which is useful in the inseparable case. However, the standard case is sufficiently important in most applications to warrant the classical treatment which I have chosen here.
Since Artin taught me algebra, my indebtedness to him is all-pervasive. It is perhaps least in the section on linear algebra and representations, where the influence of Bourbaki is more decisive (in content, not expository style). However, in choice of subject matter, I am more selective than Bourbaki, with the resulting advantages and disadvantages of being less encyclopaedic.
Granting the material which under no circumstances can be omitted from a basic course, there exist several options for leading the course in various directions. It is impossible to treat all of them with the same degree of thoroughness. The precise point at which one is willing to stop in any given direction will depend on time, place, and mood. The chapters on real fields and absolute values, for instance, can be omitted safely, or can be read by students independently of the class. The chapter on group representations also. The Witt theorem on quadratic forms can also be omitted. However, any book with the aims of the present one must include a choice of these topics, pushing ahead in deeper waters, while stopping short of full involvement, and keeping the number of pages within reasonable bounds. There can be no universal agreement on these matters, not even between the author and himself. Thus the concrete decisions as to what to include and what not to include are finally taken on grounds of general coherence and aesthetic balance. For instance, I have deliberately avoided getting too involved in commutative algebra. I could not write a basic course in algebra as an exclusive training ground for future algebraic geometers. However, anyone teaching the course will want to impress his own personality on the material, and may push certain topics with more vigour than I have, at the expense of others. Nothing in the present book is meant to inhibit this.
The order of the book is still remarkably like that given by Artin-Noether-Van der Waerden some thirty years ago. I agree wholeheartedly with Van der Waerden's inclusion of the representation theory of finite groups in a basic text. In view of progress made by Brauer during the past thirty years, it has been possible to give a much more complete treatment than Van der Waerden could at that time.
There is some reason to include more on linear groups and their representations than I could do and still have a reasonably sized book. This can be done especially with students who have a proper background in linear algebra from their undergraduate days. Fortunately, several texts dealing with Lie algebras and Lie groups are now becoming available, so that I did not feel too guilty in omitting these topics. (Cf. in particular Serre's notes, Lie Algebras and Lie Groups.)
As prerequisites, I assume only that the reader is acquainted with the basic language of mathematics (i.e. essentially sets and mappings), and the integers and rational numbers. A more specific description of what is assumed will be summarized below. On a few occasions, we use determinants before treating these formally in the text. Most readers will already be acquainted with determinants, and we feel it is better for the organization of the whole book to allow ourselves such minor deviations from a total ordering of the logic involved.
Serge Lang
New York, 1965
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