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Unformatted text preview: 9.5. THE GALOIS CORRESPONDENCE 437 determined by .˛/ , which is necessarily a root of f.x/ . But by Propo- sition 9.4.1 , there is a 2 Aut K .L/ such that .˛/ D .˛/ ; therefore, D j M . Now, the homomorphism theorem for groups gives Aut K .M/ Š Aut K .L/= Aut M .L/: This completes the proof of part (b). n We shall require the following variant of Proposition 9.5.3 . Proposition 9.5.5. Let L be a field, H a finite subgroup of Aut .L/ , and F D Fix .H/ . Then (a) L is a finite–dimensional Galois field extension of F . (b) H D Aut F .L/ and dim F .L/ D j H j . Proof. We cannot apply Proposition 9.5.3 because it is not given that L is finite–dimensional over F . Let ˇ be any element of L . We can adapt the argument of Proposition 9.4.13 to show that ˇ is algebraic and separable over F , and that the minimal polynomial for ˇ over F splits in L . Namely, let ˇ D ˇ 1 ;:::;ˇ r be the distinct elements of f .ˇ/ W 2 H g . Define g.x/ D .x ˇ 1 /.:::/.x ˇ r / 2 LOExŁ . Every 2 H leaves...
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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