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Unformatted text preview: 7.5. SPLITTING FIELDS OF POLYNOMIALS IN C OExŁ 343 intermediate field is the fixed field of a subgroup of G . The three sub groups of D 4 of index 2 ( V , V , and Z 4 ) are normal, so the corresponding intermediate fields are Galois over the ground field Q . It is possible to determine each fixed field rather explicitly. In order to do so, we can find one or more elements that are fixed by the subgroup and that generate a field of the proper dimension. For example, the fixed field of V is Q . p 2/ . You are asked in the Exercises to identify all of the intermediate fields as explicitly as possible. Finally, I want to mention one more useful result, whose proof is also deferred to Section 9.5 . Theorem 7.5.15. If K F C are fields and dim K .F/ is finite, then there is an ˛ 2 F such that F D K.˛/ . It is rather easy to show that a finite–dimensional field extension is algebraic; that is, every element in the extension field is algebraic over the ground field K . Hence the extension field is certainly obtained by....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Polynomials

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