### 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 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*.

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