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.
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
qqqqq qqqqq qqqqqqqq
qqqqq qqqqq Â Â M; p.x/ L qqqqq
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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