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College Algebra Exam Review 418

# College Algebra Exam Review 418 - 428 9 FIELD EXTENSIONS...

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Unformatted text preview: 428 9. FIELD EXTENSIONS – SECOND LOOK Aut.L/. If K Â L is a ﬁeld extension, we denote the set of automorphisms of L that leave each element of K ﬁxed by AutK .L/; we call such automorphisms K -automorphisms of L. Recall from Exercise 7.4.4 that AutK .L/ is a subgroup of Aut.L/. Proposition 9.4.1. Let f .x/ 2 KŒx, let L be a splitting ﬁeld for f .x/, let p.x/ be an irreducible factor of f .x/, and ﬁnally let ˛ and ˇ be two roots of p.x/ in L. Then there is an automorphism 2 AutK .L/ such that .˛/ D ˇ . Proof. Using Proposition 9.2.1, we get an isomorphism from K.˛/ onto K.ˇ/ that sends ˛ to ˇ and ﬁxes K pointwise. Now, applying Corollary 9.2.5 to K.˛/ Â L and K.ˇ/ Â L gives the result. I Proposition 9.4.2. Let L be a splitting ﬁeld for p.x/ 2 KŒx, let M; M 0 be intermediate ﬁelds, K Â M Â L, K Â M 0 Â L, and let be an isomorphism of M onto M 0 that leaves K pointwise ﬁxed. Then extends to a K -automorphism of L. Proof. This follows from applying Proposition 9.2.4 to the situation speciﬁed in the following diagram: L ppppppppppppppppp qqqqq qqqq qqq qqqqq qqqqqqqq qqqqq qqqqq qqqqqqqq qqqqq qqqqq Â Â M; p.x/ L qqqqq qqqq qqq qqqqq M 0 ; p.x/ I Corollary 9.4.3. Let L be a splitting ﬁeld for p.x/ 2 KŒx, and let M be an intermediate ﬁeld, K Â M Â L. Write IsoK .M; L/ for the set of ﬁeld isomorphisms of M into L that leave K ﬁxed pointwise. (a) There is a bijection from the set of left cosets of AutM .L/ in AutK .L/ onto IsoK .M; L/. ...
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