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Let  be a field and  its extension field. If
 , then the smallest subfield of  , that contains  and  , is denoted by  . We say that  is obtained from the field  by adjoining the element  to  via field adjunction.
Proof. (1) Because ![$ K[\alpha]$](http://images.planetmath.org:8080/cache/objects/5875/l2h/img15.png "$ K[\alpha]$") is an integral domain (as a subring of the field  ), all possible quotients of the elements of ![$ K[\alpha]$](http://images.planetmath.org:8080/cache/objects/5875/l2h/img17.png "$ K[\alpha]$") belong to  . So we have
and because  was the smallest, then
(2)  is a subring of  containing  and  , therefore also the whole ring ![$ K[\alpha]$](http://images.planetmath.org:8080/cache/objects/5875/l2h/img26.png "$ K[\alpha]$") , that is,
![$ K[\alpha] \subseteq K(\alpha)$](http://images.planetmath.org:8080/cache/objects/5875/l2h/img27.png "$ K[\alpha] \subseteq K(\alpha)$") . And because  is a field, it must contain all possible quotients of the elements of ![$ K[\alpha]$](http://images.planetmath.org:8080/cache/objects/5875/l2h/img29.png "$ K[\alpha]$") , i.e.,
 .
In addition to the adjunction of one single element, we can adjoin to  an arbitrary subset  of  : the resulting field  is the smallest of such subfields of  , i.e. the intersection of
such subfields of  , that contain both  and  . We say that  is obtained from  by adjoining the set  to it. Naturally,
The field  contains all elements of  and  , and being a field, also all such elements that can be formed via addition, subtraction, multiplication and division from the elements of  and
 . But such elements constitute a field, which therefore must be equal with  . Accordingly, we have the
Theorem 2  constitutes of all rational expressions formed of the elements of the field  with the elements of the set  .
Notes.
1.  is the union of all fields  where  is a finite subset of  .
2.
 .
3. If, especially,  also is a subfield of  , then one may denote  .
- 1
- B. L. VAN DER WAERDEN: Algebra. Erster Teil. Siebte Auflage der Modernen Algebra. Springer-Verlag; Berlin, Heidelberg, New York (1966).
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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
![$ K[\alpha]$](http://images.planetmath.org:8080/cache/objects/5875/l2h/img14.png "$ K[\alpha]$"){kind=small}
![$\displaystyle K\cup\{\alpha\} \subseteq K[\alpha] \subseteq Q \subseteq E,$](http://images.planetmath.org:8080/cache/objects/5875/l2h/img19.png "$\displaystyle K\cup\{\alpha\} \subseteq K[\alpha] \subseteq Q \subseteq E,$"){kind=small}
