Getallenveld

Getallenveld, in Dutch mathematics, is the term for a field whose elements are numbers, equipped with two operations: addition and multiplication. A field is a set in which these operations satisfy the usual axioms: associativity, commutativity, distributivity, the existence of additive and multiplicative identities, and inverses for every nonzero element. In a getallenveld, 0 and 1 are distinct elements, and division by a nonzero element is always possible.

In a narrower, standard mathematical sense, a getallenveld is a number field: a field K that is a finite extension of the rational numbers Q. This means Q is contained in K and the dimension of K as a vector field over Q, denoted [K:Q], is finite. Such fields can be constructed by adjoining a root of a polynomial with rational coefficients, and every element of K can be expressed as a finite combination of that root with rational coefficients.

Examples range from familiar number systems to more advanced constructs. The rational numbers Q, the real numbers R, and the complex numbers C are fields that are used as bases for arithmetic and analysis; R and C are not finite extensions of Q, but they are common getallenvelden in broader contexts. In the finite-field setting, fields such as GF(p) and GF(p^n) (p a prime) are fields of numbers used extensively in algebra, number theory and coding theory, though they are not number fields in the strict sense of finite extensions of Q.

Key properties include a characteristic of zero in most number-field contexts, the presence of Q as a subfield, and, for finite extensions, a finite degree [K:Q]. Number fields are central in algebraic number theory and play a major role in arithmetic geometry and related areas.