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.

No comments:

Post a Comment

Our Latest Post

Introduction of Circle

All of my lessons and teaching videos are in English and most of them are for students of Logistics Management. But many of mys students an...

Popular Post