minimalfield

Minimalfield is a term used in field theory to describe the smallest subfield of a given field that contains a specified set of elements. More formally, if F is a field, K is a subfield of F, and S is a subset of F, the minimal field containing K and S is denoted K(S). It is defined as the intersection of all subfields of F that contain K ∪ S, and it can be viewed as the field obtained by adjoining the elements of S to K.

A key special case is the prime field. The prime field of a field F is the smallest subfield of F, obtained by adjoining 1 to the prime subfield. If the characteristic of F is a prime p, the prime field is isomorphic to the finite field Fp. If the characteristic is zero, the prime field is isomorphic to the rational numbers Q. Every field contains its prime field, and any larger subfield is built by adjoining more elements to this base.

Construction and examples. The field K(S) can be constructed by closing K ∪ S under addition, multiplication, and taking inverses. For example, in the field of complex numbers C, the field generated by Q and i is Q(i), the Gaussian rationals, consisting of a + bi with a, b ∈ Q. In general, K(S) is finite when S is algebraic over K, and has positive transcendence degree when S contains transcendental elements.

Notes. The concept is central to definitions of function fields, algebraic closures, and field extensions. It generalizes the intuitive idea of “adjoining” elements to a base field to obtain the smallest subfield containing given data.