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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
a condition of algebraic extension
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(Theorem)
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Theorem. A field extension  is algebraic if and only if any subring of the extension field  containing the base field  is a field.
Proof. Assume first that  is algebraic. Let  be a subring of  containing  . For any non-zero element  of  , naturally
![$ K[r] \subseteq R$](http://images.planetmath.org:8080/cache/objects/10379/l2h/img10.png "$ K[r] \subseteq R$") , and since  is an algebraic element over  , the ring ![$ K[r]$](http://images.planetmath.org:8080/cache/objects/10379/l2h/img13.png "$ K[r]$") coincides with the field  . Therefore we have
![$ r^{-1} \in K[r] \subseteq R$](http://images.planetmath.org:8080/cache/objects/10379/l2h/img15.png "$ r^{-1} \in K[r] \subseteq R$") , and  must be a field.
Assume then that each subring of  which contains  is a field. Let  be any non-zero element of  . Accordingly, the subring ![$ K[a]$](http://images.planetmath.org:8080/cache/objects/10379/l2h/img21.png "$ K[a]$") of  contains  and is a field.
So we have
![$ a^{-1} \in K[a]$](http://images.planetmath.org:8080/cache/objects/10379/l2h/img24.png "$ a^{-1} \in K[a]$") . This means that there is a polynomial  in the polynomial ring ![$ K[x]$](http://images.planetmath.org:8080/cache/objects/10379/l2h/img26.png "$ K[x]$") such that
 . Because
 , the element  is a zero of the polynomial  of ![$ K[x]$](http://images.planetmath.org:8080/cache/objects/10379/l2h/img31.png "$ K[x]$") , i.e. is algebraic over  . Thus every element of  is algebraic over  .
- 1
- DAVID M. BURTON: A first course in rings and ideals. Addison-Wesley Publishing Company. Reading, Menlo Park, London, Don Mills (1970).
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"a condition of algebraic extension" is owned by pahio.
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(view preamble)
Cross-references: zero of the polynomial, polynomial ring, polynomial, contains, ring, algebraic, proof, field, base field, extension field, subring, field extension
There is 1 reference to this entry.
This is version 4 of a condition of algebraic extension, born on 2008-03-08, modified 2008-03-08.
Object id is 10379, canonical name is AConditionOfAlgebraicExtension.
Accessed 231 times total.
Classification:
| AMS MSC: | 12F05 (Field theory and polynomials :: Field extensions :: Algebraic extensions) |
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Pending Errata and Addenda
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