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A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers.
Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.)
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules.
Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. [1] [2] The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order.
In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension / has finite ...
In mathematics, particularly in algebra, a field extension is a pair of fields, such that the operations of K are those of L restricted to K. In this case, L is an extension field of K and K is a subfield of L .
A global field is one of the following: An algebraic number field. An algebraic number field F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The function field of an irreducible algebraic curve over a ...
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. [2] [3] [4] This means the following: The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real ...