Preface to the Advanced Algebra Learning Instruction Book (third edition) of Fudan University

The second edition of the book, since its publication in 2007, has received the attention and affirmation of the vast number of readers. In the course of 8 years ' teaching practice, we have received various opinions and suggestions from our brothers and universities ' peers and students. In addition, the general higher education "Twelve-Five" National planning textbook "Advanced Generation of Mathematics (third edition)" (Fudan University Press) has been formally published in October 2014, so as the teaching materials supporting the study method guidance book, the third edition of this book revision work also put on the agenda.

The third edition of the book maintains the original framework and system of the second edition, but has made further modifications and improvements in the following areas. First of all, corrected the second version of the errors and irregularities, and the original subject classification of the appropriate adjustments to make the hierarchy more distinct. Second, each chapter has added a large number of examples, so that the total number of examples reached more than 700, the length of the second version of the increase of nearly two-thirds, the book content richer. Finally, on the basis of the second edition, we also add the content that is not involved in the textbook of similar standard type theory, linear mapping and generalized inverse of matrices based on elementary factors in general number field.

Different from the general higher algebra problem Sets or postgraduate study guidance books, the book has the following characteristics:

First, the book is written in the order of chapters and "Advanced Generation of Mathematics (third edition)" textbook exactly the same, which enables beginners to learn the textbooks and guidance in a synchronized manner, gradually master the various types of knowledge points in advanced algebra and the problem-solving methods.

Second, in the topic Division and the selection of examples are mainly based on methods and techniques to classify, rather than the examples in accordance with the chapter content simply listed together, highlighting the methods and techniques in the higher algebra learning process play a role.

Thirdly, in the beginning of each chapter or in the comments after some examples, the problem solving methods and techniques of the same type of problems will be summed up and summed up, which allows the reader through the typical example of learning can extrapolate, comprehend by analogy, avoid trapped in stuffed.

Four, the book more than 100 typical examples give a variety of evidence method or solution, these different methods are often interspersed in the chapters, and with the depth of the course is constantly excavated. This not only reflects the organic connection between the various knowledge points of higher algebra, but also greatly stimulates the reader's divergent thinking and improves the reader's ability to solve problems.

Five, each chapter of the structure is first elaborated the basic concepts and theorems, and then carefully explain the typical examples, and finally arranged a number of basic training problems (the book a total of 450), including single-choice, fill-in questions and 3 types of reconciliation answer. This order conforms to the law of learning, and is also the usual teaching method of higher algebra exercises in the College of Mathematics Science of Fudan University, which has been affirmed by successive students and has achieved good teaching effect.

The third edition of the book incorporates the editors ' experience of teaching advanced algebra at the School of Mathematical Sciences at Fudan University and the latest teaching results (see [8]). The book selected examples of rich, difficult, most from the College of Mathematical Science, Fudan University of higher algebra, final exam questions (including some original propositions), there are some examples selected from the Brothers College graduate Entrance Examination questions and the National Mathematics Competition test.

This book, by explaining examples, gives a complete summary of the related properties of many important points of knowledge, such as the nature of adjoint matrices. In addition, many important methods of solving the application, although scattered in the chapters, but as long as the reader can still be linked to the whole reading, such as the application of the perturbation method. Therefore, the book is also suitable for students to participate in the postgraduate entrance examination or participate in the National Mathematics competition for students to prepare for the exam.

In the editor's view, the method of learning advanced algebra should be "deep understanding of geometric meaning;" Proficiency in algebraic methods; Emphasizing the mutual transformation and organic unification of algebra and geometry, which is the guiding ideology of this book. Readers are expected to learn and master Advanced algebra by reference to this book.

This book has been prepared by the National Natural Science Foundation (No. 11422101) and the School of Mathematics Science, Fudan University, received the help of Fudan University Press Fan Renmei, and expressed heartfelt thanks! Sincerely welcome the broad masses of readers and peer experts to make further criticisms and suggestions.

Liang Sheiqi

July 2015 at Fudan University

Fudan University Advanced Algebra Study Guide (third edition) preface