Let be a field and an extension field of . 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.
and because was the smallest, then
(2) is a subring of containing and , therefore also the whole ring , that is, . And because is a field, it must contain all possible quotients of the elements of , i.e., .
In 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. constitutes of all rational expressions formed of the elements of the field with the elements of the set .
1. is the union of all fields where is a finite subset of .
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).
|Date of creation||2015-02-21 15:39:45|
|Last modified on||2015-02-21 15:39:45|
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