15/05/05 16:00:51.22 8svvg2D/.net
>>302 つづき
P140より
THEOREM 7. A ny automorphism of a subfield of C can be extended to an automorphism of C.
Proof. Let φ be an automorphism of a subfield of C, and let F(ヒゲ) = {θl θ is an
automorphism extending φ to some subfield of C}. The proof that F(ヒゲ) satisfies
the three hypotheses of Zorn's lemma is virtually the same as in the proof of
Theorem 6, the only change necessary is to show that domain σ = range σ instead
of domain σ⊆F^α. We leave this to the reader. Applying Zorn's lemma let
ψ be a maximal member of F(ヒゲ). We must show domain ψ = C. If not, then there
is a complex number, α, not in domain ψ=F. If α is algebraic over F then, by
Theorem 6, we could extend ψ to an automorphism of F^α contradicting the
maximality of ψ in F(ヒゲ). If a is transcendental over F, then by Theorem 5B we
could extend ψ to an automorphism of F(a), sending α to α for example, since
α is also transcendental over range ψ = F. This again contradicts the maximality
of ψ, so there can be no complex numbers outside of domain ψ and the proof is complete.
引用おわり
”the only change necessary is to show that domain σ = range σ instead
of domain σ⊆F^α. We leave this to the reader. ”と言われてしまった・・
Zorn's lemma は、使い易いのかね？ よく出てくる・・
追伸
余談だが、F(ヒゲ)では高等数学の雰囲気が壊れるが、ヒゲ文字をこの貧弱な板で簡単に出す方法を知らないのでご勘