Unformatted text preview: D . Because g.x/ splits into linear factors in EŒxŁ , so does h.x/ . Thus E is a splitting ﬁeld for h.x/ . Since deg .h/ D dim M .E/ µ 2 , and since the roots of h.x/ are distinct by Exercise 7.4.1 , h.x/ has at least one root b 2 E other than a and, by Proposition 7.4.2 , there is an M –automorphism of E that takes a to b . Therefore, Aut M .E/ ¤ f e g : n Proposition 7.4.8. Let K ² M ² E be an intermediate ﬁeld. Let H D Aut M .E/ and let M D Fix .H/ . (a) If H is one of the H i , then M D M D K.˛ i / . (b) If H D A 3 , then M D M D K.ı/ . (c) If H D Aut K .E/ , then M D M D K . (d) If H D f e g , then M D M D E . Proof. If H D Aut K .E/ Š S 3 , then M D M D K , by Proposition 7.4.6 . If H D f e g , then M D M D E , by Proposition 7.4.7 . To complete...
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 Fall '08
 EVERAGE
 Algebra, Cyclic group, AutK .E/, AutM .E/

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