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Algebraic Numbers and Fields

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Let be fields, and algebraic over F. show that if and only if for some .
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Let be fields, and algebraic over F. show that if and only if for some .

Solution:

Let  ,   K . Since K : F is separable ,  and  have distinct conjugates {  =  1 , .... ,  m } and {  =  1 , .... ,  n } , respectively , where the minimal polynomial of each over F has degree m and n , respectively . We assume that m , n  2 , since otherwise it immediately follows as ...

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