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1.^ J. S. Milne, Fields and Galois Theory, pp.100-101.（上記に同じ）
P113
REMARK 9.18 What are the automorphisms of C? There are only two continuous automorphisms
(cf. Exercise A-8 and solution). If we assume Zorn’s lemma, then it is easy to
construct many: choose any transcendence basis A for C over Q, and choose any permutation
α of A; then α defines an isomorphism Q(A)→Q(A) that can be extended to an
automorphism of C. Without Zorn’s lemma, there are only two, because the noncontinuous
automorphisms are nonmeasurable,*1 and it is known that the Zorn’s lemma is required to
construct nonmeasurable functions.*2
9 Transcendental Extensions 109
Algebraic independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Transcendence bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Luroth’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Separating transcendence bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Transcendental Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117