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 ﬁeld, H a subgroup of Aut .L/ and K ² L a subﬁeld. Then (a) Fix .H/ is a subﬁeld of L . (b) Aut Fix .H/ .L/ ³ H . (c) Fix . Aut K .L// ³ K . Proof. Exercise 9.4.1 . n Proposition 9.4.8. Let L be a ﬁeld, H a subgroup of Aut .L/ , and K ² L a subﬁeld. Introduce the notation H ı D Fix .H/ and K D Aut K .L/ . The previous exercise showed that H ı ³ H and L ı ³ L . (a) If H 1 ± H 2 ± Aut .L/ are subgroups, then H ı 1 ³ H ı 2 . (b) If K 1 ± K 2 ± L are ﬁelds, then K 1 ³ K 2 . Proof. Exercise 9.4.2 . n Proposition 9.4.9. Let L be a ﬁeld, H a subgroup of Aut .L/ , and K ² L a subﬁeld. (a) .H ı / 0ı D H ı . (b) .K / ı0 D K ....
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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
 EVERAGE
 Algebra, Polynomials

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