Field theory (mathematics)

Ring and field axioms
	Ring	Commutative ring	Skew field or Division ring	Field
Abelian (additive) group structure	Yes	Yes	Yes	Yes
Multiplicative structure and distributivity	Yes	Yes	Yes	Yes
Commutativity of multiplication	No	Yes	No	Yes
Multiplicative inverses	No	No	Yes	Yes

A field is a fundamental algebraic structure, which is widely used in algebra, number theory and many other areas of mathematics. It is a set on which are defined addition, subtraction, multiplication, and division, such that they behave as they do when applied to rational and real numbers.

The most well known fields are the field of rational numbers, and the field of real numbers. In addition, the field of complex numbers, is widely used, not only in mathematics, but also in many areas of science and engineering. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Finite fields are used in most used for computer security.

Any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. The theory of field extensions (including Galois theory) involves the roots of polynomials with coefficients in a field; among other results, this theory leads to impossibility proofs for the classical problems of angle trisection and squaring the circle with a compass and straightedge, as well as a proof of the Abel–Ruffini theorem on the algebraic insolubility of quintic equations.