Mathematical Field
A Mathematical Field is a nonzero commutative ring that contains a multiplicative inverse for every nonzero element.
- See: Finite Field, Abstract Algebra, Affine Space, Abelian Group, Algebraic Structure, Addition, Subtraction, Multiplication, Division (Mathematics).
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/field_(mathematics) Retrieved:2015-2-7.
- In abstract algebra, a field is a nonzero commutative ring that contains a multiplicative inverse for every nonzero element, 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, fields of functions, 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.)
As a ring, a field may be classified as a specific type of integral domain, and can be characterized by the following (not exhaustive) chain of class inclusions:
: Commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields.
- In abstract algebra, a field is a nonzero commutative ring that contains a multiplicative inverse for every nonzero element, 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, fields of functions, algebraic number fields, p-adic fields, and so forth.
2013
- http://mathworld.wolfram.com/Field.html
- QUOTE: A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra. An archaic name for a field is rational domain. The French term for a field is corps and the German word is Körper, both meaning "body." A field with a finite number of members is known as a finite field or Galois field.
Because the identity condition is generally required to be different for addition and multiplication, every field must have at least two elements. Examples include the complex numbers (C), rational numbers (Q), and real numbers (R), but not the integers (Z), which form only a ring.
It has been proven by Hilbert and Weierstrass that all generalizations of the field concept to triplets of elements are equivalent to the field of complex numbers.
- QUOTE: A field is any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra. An archaic name for a field is rational domain. The French term for a field is corps and the German word is Körper, both meaning "body." A field with a finite number of members is known as a finite field or Galois field.