Overview

Late-nineteenth-century concerns about the meaning of number led, in the early twentieth century, to attempts to provide an axiomatic foundation for algebra and the theory of numbers. Building on the earlier work of Niels Abel and Evariste Galois on algebraic equations and using the framework provided by Cantor's theory of sets, a group of mathematicians led by Emmy Noether formalized definitions for a number of algebraic structures. Emphasis in mathematical research shifted from finding solutions to equations to the structures that such sets of solutions exhibit. The theory of groups, in particular, provided an important tool for theoretical physics.

Background

As mathematics has developed over the centuries, the concept of number has been broadened and generalized. Originally only the counting numbers, 1,2,3,..., or positive integers were recognized. The introduction of the zero and the negative integers turned arithmetic into a...