| Algebraic Closure |
Article Index for Algebraic |
Shopping Closure |
Website Links For Algebraic |
Information AboutAlgebraic Closure |
| CATEGORIES ABOUT ALGEBRAIC CLOSURE | |
| field theory | |
|
Using Zorn's Lemma , it can be shown that every field has an algebraic closure, and that the algebraic closure of a field ''K'' is unique Up To an Isomorphism that Fixes every member of ''K''. Because of this essential uniqueness, we often speak of ''the'' algebraic closure of ''K'', rather than ''an'' algebraic closure of ''K''. The algebraic closure of a field ''K'' can be thought of as the largest algebraic extension of ''K''. To see this, note that if ''L'' is any algebraic extension of ''K'', then the algebraic closure of ''L'' is also an algebraic closure of ''K'', and so ''L'' is contained within the algebraic closure of ''K''. The algebraic closure of ''K'' is also the smallest algebraically closed field containing ''K'', because if ''M'' is any algebraically closed field containing ''K'', then the elements of ''M'' which are algebraic over ''K'' form an algebraic closure of ''K''. The algebraic closure of a field ''K'' has the same Cardinality as ''K'' if ''K'' is infinite, and is Countably Infinite if ''K'' is finite. EXAMPLES
SEPARABLE CLOSURE An algebraic closure of ''K'' contains a subfield ''K''''s'', which contains all the finite Separable Extension s of ''K'' within it. For ''K'' a Perfect Field , the algebraic and separable closures are the same. In other cases the separable closure must be used to define the Absolute Galois Group of ''K''. |
|
|