Abstract Algebra, offered by Xiaotao Sun, Dean of School of Mathematics of Tianjin University, has been recognized as a national first-class undergraduate course.
Abstract Algebra is not only a professional course for senior students in mathematics department, but also an extension and improvement of the basic course Advanced Algebra. The design concept of the course is based on the classical mathematical problems as the guidance, according to the students to accept the concept from concrete to abstract, from familiar to unfamiliar. Students are required to master the basic concepts and theorems of group, ring and field, especially to understand their background and how to use them to describe and solve the problem of ruler drawing and the problem of whether polynomial equation has radical solution. To cultivate students' ability to appreciate mathematics and understand the ideas and methods of modern mathematics.
The course leader is Xiaotao Sun, and the team members include Jijian Song, Xun Yu, Mingshuo Zhou, etc.
Many students' impression of the course of Abstract Algebra is that they have learned a lot of basic concepts of "group, ring and field", but they don't know what they use them for. In fact, the introduction of some basic concepts in mathematics and the discussion of their properties are by-products of describing and solving some mathematical problems. If we only study these concepts without understanding their background and problems to be solved, it is just like a student of British and American literature who focuses on learning classical English vocabulary and grammar without understanding Shakespeare.
What is Shakespeare in Abstract Algebra? There is no doubt that there are some famous classical problems in the history of mathematics. The description and solution of these problems actually produced the basic concepts of "group, ring, field" in "Abstract Algebra".
As a graduate of mathematics department, no matter he is engaged in mathematics research in the future, or a mathematics teacher in primary and secondary schools, he should understand how these problems are solved and the historical story of the problems. It's also one of the most beautiful parts of mathematics.
This course will focus on the description and solution of these problems to teach the course of Abstract Algebra. Guided by classical problems, the basic concepts of "group, ring and field" are introduced in the most natural way. In order to better achieve the teaching objectives, Professor Xiaotao Sun, who is in charge of the course, has compiled his own lecture notes and teaching materials.
The design concept of this course is based on the classical mathematical problems as the guidance, according to the students to accept the concept from concrete to abstract, from familiar to unfamiliar. Around the problem of solving equations, some concepts appear repeatedly in the whole course and gradually deepen, making it easier for students to understand and accept.
According to this idea, the first class (2 class hours) introduces the students to the problem of ruler drawing and whether there are radical solutions to higher-order equations. By showing the solution formula of cubic equation to students, the meaning of "equation solvable" is explained: its solution can be obtained by finite times of "four mixed operations + square". It shows that it is reasonable for us to consider "the set closed to the four mixed operations", which lays the foundation for the definition of field. At the same time, it introduces the answers to the above questions to the students, so that they have the desire to understand "group, ring and field". In the following chapter, we introduce the basic properties of "fields, rings, groups" and their homomorphism.
The second chapter focuses on the unique decomposition of integer rings and polynomial rings, and introduces the concept of unique decomposition of integral rings. Through the study of the basic concepts in the first two chapters, students are prone to fatigue.
So in the third chapter, we first show the students how to use the concepts to describe and solve the problem of ruler drawing. At the same time, it asserts that the problem of ruler drawing is actually a special case of "whether the equation has a radical solution" to stimulate students' thirst for knowledge, and tells students that in order to strictly describe and solve the problem of "whether the equation has a radical solution", we must introduce the splitting field and study the "intermediate field" between it and the equation coefficient field. In the last section of the chapter "group and field expansion", we prove the corresponding relationship between "intermediate field" and "subgroup", which provides a motive power for the study of Chapter 4 "preliminary group theory".
In Chapter 4, section 4 "solvability of Galois group", we prove that "Galois group is solvable" if and only if "equation is solvable", and prove that general higher order equation is "insoluble". In the fifth section of the fourth chapter, we introduce the concept of "group action" to the students through the proof of Sylow theorem. Here we emphasize that "the function of a group on a set defines the equivalence relation on a set", and reinterpret the coset decomposition. The fifth chapter "preliminary model theory" is the selected content, which depends on the specific situation of students.
Yanan Lin, who was selected for the national ten thousand talents program of Xiamen University, believes that the selection and arrangement of the course content has new ideas and characteristics, which solves the regret that we all think that Abstract Algebra did not play on the stage and will soon be popularized in China. The lectures are refined and in place, which reflects the charm of mathematicians. Hope to publish a book as soon as possible, especially to write exercises reasonably, so that more teachers and students benefit.
Hourong Qin, a distinguished professor of Changjiang Scholars at Nanjing University and winner of the national fund for Distinguished Young Scholars, said: students often fall into a strange circle when they study Abstract Algebra: they are confused by a bunch of abstract concepts, and they don't understand why these abstract theories are put forward? What's the use? This phenomenon is very common and is a chronic disease in the teaching of Abstract Algebra. Abstract Algebra course by Professor Xiaotao Sun is a brand new design. It introduces rings and fields first, and then introduces group theory by using Galois theory. This design is refreshing. The sequence from field to group is also Galois's idea in that year, which is in line with the law of historical development of mathematical theory. It is helpful for students to understand the background and development process of mathematical problems, so that they can more easily understand abstract concepts and use them. This is a very useful attempt, which opens up a new way of thinking for the teaching of Abstract Algebra, and is worth learning from the domestic counterparts.