Examples of Applications

Throughout history, the theory of numbers was considered as a purely theoretical science with no practical applications and connections to the real world. The first important application of group theory was the classification of all possible structures of crystals, which is one of the basic problems in crystallography. The study of crystallographic groups was started by J. S. Fiedorow and continued by A. Schonflies at the end of the 19-th century. They showed that there are only 17 plane crystallographic groups and there are exactly 230 different three-dimensional crystallographic groups. It was the first example of the application of group theory to natural science. Presently, the application of the theory of numbers, groups and rings covers such different branches of sciences and technics as cryptography and quantum mechanics, generics and particle physics, biology and chemistry, even music and painting. This chapter considers some examples of the application of the theory of numbers, groups and rings.