[Abstract Algebra] 01-mathematical "Yi tianjian"

In the general impression, mathematics is used for calculation. This is not a general case, because most mathematical applications are used to obtain a value. However, from the perspective of mathematical objects, computation is sometimes not the main character. The simplest example is the plane ry that everyone is familiar with. It is often only studying the "relationship" between points and lines ". At the beginning, Dai math was used as a symbolic representation of computation. However, as its scope of use expands, it is found that it can also represent various "relationships ". In the set theory, we have seen the exact definition of the "relationship", so I will discuss it in depth here. The relationship exists in many application models, and they have a very similar structure in nature,Abstract AlgebraIt is the science of studying these structures.

The story has to start with solving the equation. When talking about the quadratic equation of a single element, we certainly won't forget the Weida theorem. Verda first introduced concise algebraic symbols into the expression of the formula, and gave the algebraic expression of the solution of the second to four equations. You may not know that the solutions to any one-dimensional cubic or four-time equation actually have a complete formula. They were completely solved as early as the end of the Middle Ages in Europe. But the strange thing is that people cannot find the root solution of the five or more equations of a single element. Just like the "three major difficulties in plotting", at the beginning, we thought that we did not find the correct method, but never thought that they could not be done!

The solution to the five equations has dragged on to the 19th century, where he first discovered the key role of replacement in the solution of the equation. The young Abel demonstrated the following path: there is no general root solution for the five equations. Galova, who is also young, goes further. It first puts forward the concept of a group, thoroughly gives sufficient and necessary conditions for the root-type solution of the equation, and completely solves this problem. The introduction of group and galova's theory indicate the birth of abstract algebra. It will replace calculus with overwhelming advantages and become the pillar of mathematics. It also serves the watershed of modern mathematics and traditional mathematics. Abstract Algebra is the omnipotent key in mathematics. Its Introduction almost changes the face of all mathematical branches. It re-opens the analysis, ry, number theory, topology and other disciplines from a new perspective. Of course, the full establishment of abstract algebra has taken a long time and has been built by countless excellent mathematicians. You can find its historical overview on the Internet, which is not mentioned here.

Galois (1811-1832)

Abstract Algebra is now a basic mathematical method. It not only plays a pillar role in pure mathematics, but also has a universal value, it is useful in physics, crystallization, and cryptography. This blog is only intended to introduce basic concepts and conclusions, as well as advanced content plans. I feel that my current understanding is still superficial, and I still need to constantly understand the essence. I personally think that abstract algebra is a philosophy in mathematics. It is the most valuable thing to constantly think about its nature. It is also of great help to the exercise of abstract thinking and the establishment of scientific methodologies.

 

[Preface] Set Theory, elementary Number Theory

[Reference Materials] (continuously updated)

[] Modern algebra, Yang zixiao, 2000

A good domestic textbook with comprehensive content and clear structure. Exercises are enlightening, but some prove that there is no train of thought, and it looks hard.

[] Modern algebra (2rd), Han Shian, 2009

The subject content is relatively simple, and some simple examples are explained in detail, suitable for beginners. The additional content is in-depth and the exercises are rich. There are many historical stories in the book, which can increase your interest in reading.

[] Introduction to modern algebra, Feng keqin, 2002

The content is rough and compact, rich and clear, and can be used as an advanced reader.

[] Introduction to substitute Mathematics (2nd), Ling Linshao, 2009

This paper introduces the important conclusions of abstract algebra, and the content is simplified.

[] Abstract Algebra basics, Li kezheng, 2007

This article gives a very compact introduction to the important conclusions of abstract algebra without detailed proof. However, it provides many examples to give readers a perceptual knowledge of abstract algebra. The Advanced content is very deep and the reading materials are deep. It is suitable for advanced reading.

[] Abstract Algebra overview, Cao xihua, 1990

Abstract Algebra is an overview of the main issues and conclusions. It is an academic introduction and introduces more cutting-edge issues.

[] Question three hundred of modern algebra, Feng keqin, 2010

A good exercise set, which is difficult and widely covered, deserves careful consideration.

[] Modern algebra exercises, Yang zixiao, 2003

A large number of questions, detailed explanation, most of the content is relatively basic, suitable for beginners.

[Abstract Algebra] 01-mathematical "Yi tianjian"