A Research Of The Algebra Type Theory

Algebra form is an important, basic concept in mathematics. The algebraic invariant is an important subject in mathematical field; and it is an important tool to other fields. Based on prior to researches, this paper completely analyzes the development of algebraic invariant, followed its development time.First, the mathematical idea in the budding of algebraic invariant was discussed in detailed. The appearance of the quaternion number gives rise to the idea of linear transformation. From the quaternion to n-nary, to the concept of n-dimensional space introduced by Gloria, the bi-quaternion was given; the emergence of the idea of matrix promoted the development of linear algebra. Transformation has become a basic method in the mathematical field, including numerical theory, algebra and geometry. Hence, it is an inevitable thing that the transformation became a theory in 19^th century. Thus it was the mathematician's work to develop the algebraic invariant. However, at that time, the concept of algebraic invariant was only a tool to solve realistic problems. So, it was not studied as a concept. From the late 18^th century to the early 19^th century, the algebraic invariant was triggered by the quadrant form of numerical theory, projective geometry and the linear transformation of differential geometry.Second, revealed the birth of algebraic invariant and the great mathematicians' algebraic invariant theories. In this paper, the idea of matrix was given by discussing many mathematicians' works in quadrant form, differential equation and determinant, followed the social and cultural background. The paper mainly focused on the works of Sylvester and Gloria. There was the idea about algebraic invariant in 19^th century, which was originated 18^th century. Sylvester was the first man who used the word, "algebraic invariant". He introduced some concepts about algebraic invariant and gave some important conclusion and theories about algebraic invariant. Gloria, who was the founder to the theory of algebraic invariant, studied the algebraic invariant independent of determinant and equation systems. Sylvester and Gloria structured the base of algebraic invariant together by developing the theories of algebraic invariant.Third, the late development of the theory of algebraic invariant and its maturity are inspected. In late 19^th century, mathematicians tried to find the complete base of the invariant, i.e. algebraic base. Geer Dan proved the existence of finite base to binary form. Hilbert identified the basic idea and method of Geer Dan. Hilbert's works directed the direction to study the invariant theory, and promoted the development of this theory. Hilbert's algebraic invariant theory mainly stressed the model, ring and field. Based on Hilbert's works, the school of abstract algebra, Amy Note was its representative, was born; and such that the history after the theory of algebraic invariant belongs to the history of modern abstract algebra.Last, expounded the application of algebraic invariant in physics and geometry. The history after the theory of algebraic invariant belongs to the history of modern abstract algebra. Hilbert gave prominence to the model, ring and field. Hilbert proved that every model had a base which was consisted by finite polynomials; or every idea of every polynomial field containing n-nary had a finite base, if every idea of the coefficient field of polynomial had a finite base. From 1911 to 1919, Emmy Noether wrote a lot of papers for the different situations of finite bases with Hilbert's methods and her own methods. In 20^th century, the view of abstract algebra is in the dominant position.