[過去ﾛｸﾞ] 現代数学の系譜11 ガロア理論を読む13 [転載禁止]©2ch.net (654ﾚｽ)
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128(1): 現代数学の系譜11 ガロア理論を読む [] 2015/04/17(金) 23:09:31.54 ID:N+SZ0AF4(1/5) AAS
>>125-127
どうも。スレ主です。
おっちゃん、証明ありがとう（ちょっと納得できないところもあるが）
出題者さんも、ありがとう（ちょっと納得できないところもあるが）
129(3): 現代数学の系譜11 ガロア理論を読む [] 2015/04/17(金) 23:17:07.07 ID:N+SZ0AF4(2/5) AAS
英文だが
http://math.stackexchange.com/questions/103177/why-does-mathbbc-have-transcendence-degree-mathfrakc-over-mathbbq
Why does C have transcendence degree c over Q? Jan 28 '12 Mathematics Stack Exchange
7
(Of course I assume the Axiom of Choice...) Choose a transcendence basis X={xi}i∈I for C over Q. Then C is an algebraic extension of Q(X). Now here are two rather straightforward facts:
1: If F is any infinite field and K/F is an algebraic extension, then #K=#F.
2: For any infinite field F and purely transcendental extension F(X), we have #F(X)=max(#F,#X).
Putting these together we find
c=#C=#Q(X)=max(?0,#X).
Since c>?0, we conclude c=#X.
130: 現代数学の系譜11 ガロア理論を読む [] 2015/04/17(金) 23:21:08.40 ID:N+SZ0AF4(3/5) AAS
>>129
文字化けしたか。修正下記
c=#C=#Q(X)=max(?0,#X).→c=#C=#Q(X)=max(アレフ0,#X).
Since c>?0, we conclude c=#X.→Since c>アレフ0, we conclude c=#X.
131: 現代数学の系譜11 ガロア理論を読む [] 2015/04/17(金) 23:26:09.71 ID:N+SZ0AF4(4/5) AAS
似た話で下記がある
http://mathoverflow.net/questions/15424/what-is-the-transcendence-degree-of-q-p-and-c-over-q
What is the transcendence degree of Q_p and C over Q? Feb 16 '10 MathOverflow
7
In both cases the transcendence degree is the cardinality of the continuum. CH is not needed.
This is a corollary of the following result: let K be any infinite field, and let L/K be any extension. Then
#L=max(#K,trdegKL).
To prove this, in turn it suffices to establish the following two results (each of which is straightforward):
1) If K is infinite and L/K is algebraic, then #L=#K.
2) If K is any infinite field, T={ti}i∈I is an arbitrary set of indeterminates and K(T) is a purely transcendental function field in the indeterminates T, then #K(T)?#T+#K.
http://math.stackexchange.com/questions/223043/what-is-the-cardinality-of-a-transcendence-basis-of-mathbbc-over-mathbbq
What is the cardinality of a transcendence basis of C over Q? Oct 28 '12 Mathematics Stack Exchange
132(1): 現代数学の系譜11 ガロア理論を読む [] 2015/04/17(金) 23:50:04.48 ID:N+SZ0AF4(5/5) AAS
>>129
" For any infinite field F and purely transcendental extension F(X), we have #F(X)=max(#F,#X)."がstraightforward？
purely transcendental extensionの場合はそうなる？
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