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Unformatted text preview: 336 7. FIELD EXTENSIONS FIRST LOOK 7.4.4. If F L is a field extension, show that Aut F .L/ is a subgroup of Aut .L/ . 7.4.5. Suppose F L is any field extension, f.x/ 2 FOEx , and 1 ;:::; r are the distinct roots of f.x/ in L . Prove the following statements. (a) If is an automorphism of L that leaves F fixed pointwise, then jf 1 ;:::; r g is a permutation of f 1 ;:::; r g . (b) 7! jf 1 ;:::; r g is a homomorphism of Aut F .L/ into the group of permutations Sym . f 1 ;:::; r g / . (c) If L is a splitting field of f.x/ , L D K. 1 ;:::; r / , then the homomorphism 7! jf 1 ;:::; r g is injective. 7.4.6. Let f.x/ 2 KOEx be an irreducible cubic polynomial, with splitting field E . Let H be a subgroup of Aut K .E/ . Show that Fix .H/ is a field intermediate between K and E . In the following two exercises, suppose K E be the splitting field of an irreducible cubic polynomial in KOEx and that dim K .E/ D 6 ....
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- Fall '08