Adjunction (field theory)
Let E be a field extension of a field F. Given a set of elements A in the larger field E we denote by F(A) the smallest subextension which contains the elements of A. We say F(A) is constructed by adjunction of the elements A to F or generated by A.
A finitely generated extension may not be a finite extension. For example, Q(e) is a finitely generated extension of the rationals, but of infinite degree since e is transcendental. In a sense, a finitely generated extension is a transcendental generalization of a finite extension since, if the generators in A are all algebraic, then F(A) is a finite extension of F. Because of this, most examples come from algebraic geometry.
F(A) consists of all those elements of E that can be constructed using a finite number of field operations +, -, *, / applied to elements from F and A. For this reason F(A) is sometimes called the field of rational expressions in F and A.
- Given a field extension E/F then F(Ø) = F and F(E) = E.
- The complex numbers are constructed by adjunction of the imaginary unit to the real numbers, that is C=R(i).
Given a field extension E/F and a subset A of E, let be the family of all finite subsets of A. Then
In other words the adjunction of any set can be reduced to a union of adjunctions of finite sets.
Given a field extension E/F and two subsets N, M of E then K(M ∪ N) = (K(M))(N) = (K(N))(M). This shows that any adjunction of a finite set can be reduced to a successive adjunction of single elements.