The content is increasingly abstract, axiomatic and systematic, and the emergence of group theory is a turning point.

Methods, algebra and analysis, geometry, have been mixed and intertwined. You can see two disciplines in the following algebra, and many directions have been established and developed rapidly:

The meaning of algebra:

Elementary algebra further develops in two directions: linear equations with many unknowns; Higher order equations with higher unknowns. The development of these two directions makes algebra develop to the stage of advanced algebra. Advanced algebra, as a general term for the development of algebra to an advanced stage, includes many branches. Higher algebra offered by universities now generally includes two parts: linear algebra and polynomial algebra.

Functional analysis is a branch of studying the mapping from topological linear space to topological linear space satisfying various topological and algebraic conditions. It is developed from the study of variational problems, integral equations and theoretical physics. It comprehensively uses the viewpoints of function theory, geometry and modern mathematics to study functions, operators and limit theory in infinite dimensional vector space. It can be regarded as analytic geometry and mathematical analysis of infinite dimensional vector space.

The research object of advanced algebra is further expanded on the basis of elementary algebra, and new concepts including set, vector, vector space, matrix and determinant are introduced.

These new concepts have similar operational characteristics to numbers, but their research methods and operational means are more abstract and complex. The operation of new objects is not always the basic algorithm of symbolic numbers. So algebra was brought into the algebraic system, including group theory, ring theory and domain theory. Among them, group theory is a powerful tool to study the symmetry law of mathematical and physical phenomena, and it has also become the most common and important mathematical concept in modern mathematics, and has been widely used in other disciplines.