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By the end of the 19th century a unified field of abstract algebra developed, which included classes of algebraic structures such as groups, rings, fields, and vector spaces, and their subclasses. There were many different theories, but all were unified in a few basic axioms and many theorems, and unified on a common algebraic basis. This also led to a better understanding of the relationship between various areas of mathematics. For example, the development of Galois theory led to the recognition of the importance of elliptic curves and modular forms in number theory. The theory of Lie groups and Lie algebras led to the creation of algebraic topology. The study of algebraic rings and fields led to algebraic geometry.

Another field in which abstract algebra finds applications is in the discrete mathematics of game theory. Two players use several strategies to attempt to maximize a certain mathematical object. To find the optimal strategy, a mathematician might model the players' strategies as elements of a mathematical structure and apply an abstract mathematical theory to the system of strategies. Once this is done, the mathematician searches for an optimal set of strategies.

Also, there are no longer any unifying concepts that are more fundamental than algebra. More recently, algebraic number theory, algebraic geometry, and abstract algebra have seen a revival of interest in the field of computer science. Fundamental theorems and algorithms have been developed that aid computer scientists in solving problems. In many cases, abstract algebra is used in practice to model the problem. Algebraic structures such as groups, rings, and fields are used to represent concepts that are meaningful to the programmers of computer programs.

In 1892 Segre gave a unified treatment of the solution of algebraic equations over the real and complex numbers. In 1895 Lüroth proved the theory of parametric geometry. In 1901 Zariski studied the algebraic structure of the solutions of polynomial equations over the complex numbers and showed that this structure is independent of the choice of an equation. In 1901 Waring developed the theory of homogeneous polynomials over the complex numbers. In 1904 Wedderburn showed that if an algebra has no nilpotent elements, then it is isomorphic to a matrix algebra. This result is now called Wedderburn's little theorem.

In 1887 Cohn showed that if the number field K has class number = 1, then the class group can be embedded into the Galois group of K. The case of class number = 1 for number fields is now called Hilbert's Theorem 90.[37]

The remainder of the nineteenth century saw the development of abstract algebra in several branches of mathematics, often in parallel. In 1902 Neumann studied groups with involution; he showed that a finite group with an involution is solvable, and suggested the term group with involution for what is now called a group with an involution. In 1907 Schur showed that the ring of all linear transformations of a vector space over a field of characteristic 0 is isomorphic to the ring of all upper triangular matrices. In 1905 Cayley showed that a semisimple group with a single involution is a group with involution. In 1909 Dieudonné defined a group with involution as a pair of a group and an involution that commutes. This work grew into abstract group theory. 827ec27edc