Applied Abstract Algebra

Author: Hu Guanzhang
Publisher:
Publish Date: 2000-09-01
Features: The content summary Abstract algebra (also known as modern algebra) is an important foundation of modern mathematics and has extensive applications in computer science, information science, modern physics, and modern chemistry. It is a necessary mathematical foundation for professionals engaged in modern science and technology. This book introduces the basic theory and applications of groups, rings, and fields. It is suitable for students, graduate students, and professionals in mathematics and applied mathematics, computer science, radio, physics, chemistry, biomedical engineering, and related fields.
Excerpt: Historically, there were several geometric construction problems that puzzled people for a long time. They are:
(1) The doubling the cube problem: Construct a cube whose volume is twice that of a given cube.
(2) The trisection of an arbitrary angle: Given any angle, trisect it.
(3) The squaring the circle problem: Given a circle (i.e., its radius r), construct a square whose area is equal to that of the given circle.
(4) The n-gon construction problem: Divide a circle into n equal parts. These problems were completely solved only after the emergence of modern algebra theory.
8. Solving algebraic equations by radicals
We know that any quadratic algebraic equation in one variable can be expressed in terms of radicals for its two solutions. For cubic and quartic algebraic equations, the ancients, after long efforts, also achieved this. Thus, people naturally asked: Can the roots of any algebraic equation of degree n be expressed in terms of radicals? Many efforts failed, but these efforts prompted the development of modern algebra and ultimately solved this problem.
In the early 19th century, the French young mathematician Galois (Galois) proposed the famous Galois theory while studying the solution of quintic algebraic equations, becoming a pioneer of modern algebra. However, his work was not recognized by his contemporaries, and he died prematurely at the age of 21. It was not until the late 19th century that his theory was further developed and systematically elaborated by other mathematicians.
This branch of mathematics, with a long history and filled with many interesting problems and stories, has flourished in modern times and found widespread applications. Many specialized books have emerged for specific fields, attracting an increasing number of scientists, engineers, and students to learn and master it.
Exercises
1.1.
1. How many different necklaces can be made with 5 beads using 2 colors?
2. How many essentially different ways can the vertices of a regular tetrahedron be colored with two colors?
3. How many graphs with 4 vertices exist? How many of them are non-isomorphic?
4. How can a circle be divided into 5 equal parts using a compass and straightedge?
5. How can the roots of cubic and quartic algebraic equations be expressed in radicals?
1.2. Sets and Mappings
As previously mentioned, modern algebra studies algebraic systems, which are sets equipped with one or more operations defined on them. Therefore, we must be familiar with the basic theory of sets. Since everyone has some understanding of sets and mappings from middle school, here we only review, supplement, and make some conventions.
1. Notation for sets
There are two common methods for representing sets: one is to list all elements directly, and the other is to specify the properties of the elements. For example:
A = {1, 2, 3}, S = {x | P(x)}, where P(x) represents the property of the element x.
This book frequently uses the following sets and notations:
The set of integers Z = {0, ±1, ±2, ±3, …}, the set of positive integers Z+ = {1, 2, 3, …}, the set of rational numbers Q, the set of real numbers R, the set of complex numbers C, etc.
The number of elements in a set A is denoted by |A|. If A has a finite number of elements, it is called a finite set; otherwise, it is called an infinite set.
Use |A| = ∞ to denote that A is an infinite set, and |A| < ∞ to denote that A is a finite set.