Abstract Algebra

The following sections are included:

• Preface
• About the Authors
• Contents

Group theory germinated in the research on number theory at the beginning of the 19th century. Lagrange, Raffini, and Abel used the idea of permutations of roots to study the solutions of algebraic equations. Galois found the necessary and sufficient condition for solving algebraic equations by radicals, which heralded the creation of group theory. In the past several decades, the most important achievement in group theory was the classification of finite simple groups, which was the joint work of more than 100 group theorists. Of course, the research work in group theory is far from complete. Groups are becoming more and more widely used. For example, in the second half of the 18th century, Klein applied group theory to study geometry. At the same time, Lie put forward the concept of Lie groups to solve differential equations. Group theory has many applications in other disciplines, such as physics, chemistry, etc. In this chapter, we will introduce the basic theory of groups and study the basic ideas and tools of group classification.

In this chapter, we will introduce the notion of a ring and study the fundamental properties. In view of the structure of algebraic systems, a group has only one binary operation, but a ring has two binary operations, and the two operations are related by distributive laws. Mathematical objects we routinely encounter usually have two binary operations, e.g., sets of numbers, sets of matrices, and sets of polynomials, etc. Therefore, it is important to study rings. The terminology ring was first posed by Hilbert, although many mathematicians had established a lot of beautiful results related to rings before him. Following Hilbert, Krull and Noether made great contributions to the study of rings. Ring theory was further developed by the work of Artin, Jacobson, etc. Many terminologies related to rings, such as divisor chain condition, maximal condition, and minimal condition, were first defined and studied by them. Moreover, many special rings that are studied extensively currently in ring theory were named after them (e.g., Noetherian rings, Artin rings, etc.). Ring theory is the foundation of commutative algebra, algebraic number theory and algebraic geometry. Thus, it is vital to grasp the content of this chapter for everyone who are interested in learning further algebraic courses.

In this chapter, we introduce the notion of a module and study the fundamental properties. The precise definition of a module was first given by the German mathematician Noether. It was Noether who found for the first time that the relationship between the matrix representation theory of finite groups and the theory of algebraic structures can be clarified by the notion of a module, and then carried out a series of studies. Due to this, the study of modules has always been related to the matrix representation theory and algebraic theory…

In this chapter, we will introduce the basic ideas of Galois theory and its applications in straightedge and compass construction and in solving equations by radicals. The concept of fields originated in the work of Abel and Galois on solving algebraic equations by radicals in the early 19th century. In 1871, Dedekind first introduced the concept of number fields. Ten years later, Kronecker defined rational function fields. In 1893, Weber gave the abstract definition of fields. In 1910, Steinitz studied the properties of fields and introduced the notions of prime and perfect fields. From 1928 to 1942, Artin systematically studied the relationship between groups and fields, and rewrote Galois theory with modern methods.

The following sections are included:

• Bibliography
• Index