群、环、域的理论研究与实际应用
Theoretical Research and Practical Application of Groups, Rings and Fields

群、环和域是近世代数里重要的代数系统，为了增强学习者对于它们之间的联系和应用的理解，我们做以具体的总结和深入研究。本文首先介绍群、环、域产生与发展的过程，其次绘制了具体的群环域关系分析图，然后总结了与之相关定理及命题，最后分析了群论在密码学、图形的对称变换、分子结构及物理中的应用。
Groups, rings and fields are important algebraic systems in Modern Algebra. In order to enhance learners’ understanding of the relationship and application between them, we make a specific summary and in-depth research. This paper first introduces the emergence and development of groups, rings and fields, then draws a specific analysis diagram of the relationship between groups, rings and fields, then summarizes the related theorems and propositions, and finally analyzes the application of group theory in cryptography, symmetrical transformation of graphics, molecular structure and physics.