Monday, 11 November 2013

Field Adjunctions

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 \mathcal{T} be the family of all finite subsets of A. Then
F(A) = \bigcup_{T \in \mathcal{T}} F(T).
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(MN) = (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.

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