Unformatted text preview: 9.5.1 . Let f.x/ denote the minimal polynomial of ˛ over K . M is Galois over K if, and only if, M is the splitting ﬁeld for f.x/ , by Theorem 9.4.14 . But the roots of f.x/ in L are the images of ˛ under Aut K .L/ by Proposition 9.4.4 . Therefore, M is Galois over K if, and only if, M is invariant under Aut K .L/ . If ± 2 Aut K .L/ , then ±.M/ is an intermediate ﬁeld with group Aut ±.M/ .L/ D ± Aut M .L/± ± 1 : By part (a), M D ±.M/ if, and only if, Aut M .L/ D Aut ±.M/ .L/ D ± Aut M .L/± ± 1 : Therefore, M is invariant under Aut K .L/ if, and only if, Aut M .L/ is normal. If M is invariant under Aut K .L/ , then ² W ± 7! ± j M is a homomorphism of Aut K .L/ into Aut K .M/ , with kernel Aut M .L/ . I claim that this homomorphism is surjective. In fact, an element of ± 2 Aut K .M/ is...
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 Fall '08
 EVERAGE
 Algebra, Group Theory, Normal subgroup, AutK .L/, AutM .L/

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