This preview shows page 1. Sign up to view the full content.
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 eld 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 eld 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 nor-mal. If M is invariant under Aut K .L/ , then W 7! j M is a homo-morphism 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...
View Full Document
- Fall '08