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Unformatted text preview: H is Fix .H/ D f a 2 L W .a/ D a for all 2 H g . Proposition 9.4.7. Let L be a eld, H a subgroup of Aut .L/ and K L a subeld. Then (a) Fix .H/ is a subeld of L . (b) Aut Fix .H/ .L/ H . (c) Fix . Aut K .L// K . Proof. Exercise 9.4.1 . n Proposition 9.4.8. Let L be a eld, H a subgroup of Aut .L/ , and K L a subeld. Introduce the notation H D Fix .H/ and K D Aut K .L/ . The previous exercise showed that H H and L L . (a) If H 1 H 2 Aut .L/ are subgroups, then H 1 H 2 . (b) If K 1 K 2 L are elds, then K 1 K 2 . Proof. Exercise 9.4.2 . n Proposition 9.4.9. Let L be a eld, H a subgroup of Aut .L/ , and K L a subeld. (a) .H / 0 D H . (b) .K / 0 D K ....
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 Fall '08
 EVERAGE
 Algebra, Polynomials

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