Let F b a given field K be an extension of F and a is the element of K . Suppose c is the collection o all sub field of K which contain both a and F . Evidently c is non empty as at least K itself ( containing both a and F ) belongs to it .
Denoted by F(a) the intersection o all these sub fields of K which are members of c then F(a) is also a sub fields because we know that the intersection of an arbitrary collection of sub fields of K is also a sub field of K .
The sub field F(a) contains both F and a as every member of c contains both F and a and hence by definition F(a) is an element of c .
Again F(a) being the intersection of all members of c , it therefore must be contained in every member of c .
Hence we conclude that F(a) is a sub field of K containing both F and a and itself contained in any sub field of K containing both F and a (i.e. contain every member of c ) . Therefore F(a) is the smallest sub field of K containing both F and a , an is obtained by adjoining an element a of field K of its sub field F . The above process of adjoining an element of a field to its sub field is known a Field Adjunction .
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.
If A is finite we say F(A) is finitely generated and if A consists of a single element we say F(A) is a simple extension. The primitive element theorem states a finite separable extension is simple.
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.
A subextension of a finitely generated field extension is also a finitely generated extension.
Given a field extension E/F and a subset A of E, let be the family of all finite subsets of A. Then
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.
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.