Because abstract algebra has great universality, it can often be applied to some seemingly unrelated problems. For example, some ancient problems of drawing rulers and rulers were finally solved by Galois theory, which involved domain theory and group theory. Another example of algebraic theory is linear algebra, which makes a general study of vector spaces with quantitative and directional elements.
These phenomena show that geometry and algebra, which were originally considered irrelevant, actually have a strong correlation. Combinatorial mathematics studies the method of enumerating several objects satisfying a given structure.
Structure:
Many mathematical objects, such as numbers, functions, geometry, etc., reflect the internal structure of continuous operation or the relationships defined therein. Mathematics studies the properties of these structures, for example, number theory studies how integers are represented under arithmetic operations. In addition, similar things often happen in different structures, which is further abstract.
Then it is possible to describe the state of a class of structures with axioms. What needs to be studied is to find out the structures that satisfy these axioms among all structures. Therefore, we can learn abstract systems such as groups, rings and domains. These studies (structures defined by algebraic operations) can form the field of abstract algebra.
Refer to the above content: Baidu Encyclopedia-Mathematics