College Algebra Exam Review 425

College Algebra Exam Review 425 - 9.5 THE GALOIS...

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9.5. THE GALOIS CORRESPONDENCE 435 Suppose now that K is infinite (which is always the case if the char- acteristic is zero). L is generated by finitely many separable algebraic elements over K , L D K.˛ 1 ; : : : ; ˛ s / . It suffices to show that if L D K.˛; ˇ/ , where ˛ and ˇ are separable and algebraic, then there is a such that L D K. / , for then the general statement follows by induction on s . Suppose then that L D K.˛; ˇ/ . Let K L E be a finite– dimensional field extension such that E is Galois over K . Write n D dim K L D j Iso K .L; E/ j (Corollary 9.4.17 ). Let f ' 1 D id ; ' 2 ; : : : ; ' n g be a listing of Iso K .L; E/ . I claim that there is an element k 2 K such that the elements ' j .k˛ C ˇ/ are all distinct. Suppose this for the moment and put D C ˇ . Then K. / L , but dim K .K. // D j Iso K .K. /; E/ j n D dim K L . Therefore, K. / D L . Now, to prove the claim, let p.x/ D Y 1 i<j n x.' i .˛/ ' j .˛/ C .' i .ˇ/ ' j .ˇ/ : The polynomial p.x/ is not identically zero since the ' i are distinct on K.˛; ˇ/ , so there is an element k of the infinite field K such that p.k/ ¤ 0 . But then the elements k' i .˛/ C ' i .ˇ/
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