Algebra and number theory
Algebra is the study of abstract mathematical structures that generalise well-known number systems such as the integers or the reals, and their arithmetic operations such as addition and multiplication. Some of the most important algebraic structures include rings (such as the integers), fields (such as the reals) and groups (such as the symmetries of the cube).
The study of algebra is of fundamental importance to our understanding of the natural world. The electromagnetic and nuclear forces, for example, are best understood in terms of the so-called three unitary groups SU(3), SU(2) and U(1).
The term "number theory" once referred solely to the study of the integers. However, in modern mathematics, number theorists are interested in the properties of much broader classes of numbers.
Everyday applications
While considered an area of pure mathematics, number theory is pervasive in everyday life, with prime numbers forming the key to secure e-shopping and internet banking.
Famous unsolved problems in number theory include Goldbach's conjecture - the claim that every even integer greater than two can be written as the sum of two prime numbers - and the Riemann Hypothesis, one of the million-dollar Millennium Prize Problems.