minimikenttä

Minimikenttä is a term used in Finnish algebra to denote the smallest subfield of an ambient field that contains a given subset. Formally, if F is a field and S is a subset of F, the minimikenttä of S in F is the intersection of all subfields K of F with S ⊆ K. It is often denoted ⟨S⟩F or simply ⟨S⟩ when the ambient field is understood. This construction yields a well-defined subfield of F that is minimal with respect to containing S.

Construction and basic properties: The minimikenttä ⟨S⟩F is obtained by adjoining to the prime field of F the elements of S and closing under addition, subtraction, multiplication, and taking inverses. Equivalently, ⟨S⟩F is the smallest subfield of F containing S. If S is finite, the process can be described by a finite sequence of adjoining operations; in general, it may involve infinite steps. The concept is often called the field generated by S inside F.

Examples: In the real numbers F = R with S = {√2}, the minimikenttä ⟨S⟩R equals Q(√2), the field obtained by adjoining √2 to the rational numbers. In the complex numbers F = C with S = {ω}, where ω is a primitive cube root of unity, ⟨S⟩C = Q(ω). In F = R with S = {π, e}, ⟨S⟩R = Q(π, e). These examples illustrate how the minimikenttä depends on the ambient field and the chosen generators.

Relation to extensions: For an extension E/F and S ⊆ E, ⟨S⟩E is the smallest subfield of E containing F ∪ S. The notion plays a central role in studying field towers, algebraic versus transcendental elements, and the structure of field extensions.