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e. In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
A finite field is a finite set that is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are defined and satisfy the rules of arithmetic known as the field axioms. The number of elements of a finite field is called its order or, sometimes, its size.
Noncommutative algebra. v. t. e. 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 degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over .
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard orderings. Every subfield of an ordered field is also an ordered field in the inherited order.
Field of sets. In mathematics, a field of sets is a mathematical structure consisting of a pair consisting of a set and a family of subsets of called an algebra over that contains the empty set as an element, and is closed under the operations of taking complements in finite unions, and finite intersections.
Complete field. In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the p -adic numbers).
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. [1] Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the ...
e. In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a field and satisfying the axioms implied by "vector space ...