Difference between revisions of "Field (mathematics)"

Latest revision as of 06:59, 14 September 2016

In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra.

Description

It is a nonzero commutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division satisfying the appropriate abelian group equations and distributive law.

The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, algebraic function fields, algebraic number fields, p-adic fields, and so forth.

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.

In modern mathematics, the theory of fields (or field theory) plays an essential role in number theory and algebraic geometry.

As an algebraic structure, every field is a ring, but not every ring is a field. The most important difference is that fields allow for division (though not division by zero), while a ring need not possess multiplicative inverses; for example the integers form a ring, but 2x = 1 has no solution in integers. Also, the multiplication operation in a field is required to be commutative.

A ring in which division is possible but commutativity is not assumed (such as the quaternions) is called a division ring or skew field. (Historically, division rings were sometimes referred to as fields, while fields were called commutative fields.)

See also

• Heyting field
• Lefschetz principle
• Puiseux series
• Ring
• Vector space
• Vector spaces without fields

External links

• Field (mathematics) @ Wikipedia.org