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Math Help - isomorphism extension theorem

  1. #1
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    isomorphism extension theorem

    Here's the problem:

    Let K be an algebraically closed field. Show any isomorphism \sigma of K onto a subfield of K such that K is algebraic over \sigma[K] is an automorphism of K, that is show \sigma[K]=K.

    I know \sigma^{-1}:\sigma[K] \rightarrow K can be extended to an isomorphism \mu:K\rightarrow K' where K'<=K. And since K<=K'<=K we know \sigma^{-1} can only be extended to an automorphism of K. But does this help me? I don't see how to make the connection with \sigma[K] .

    Any advice would be great! :-)
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  2. #2
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    Quote Originally Posted by ziggychick View Post
    Here's the problem:

    Let K be an algebraically closed field. Show any isomorphism \sigma of K onto a subfield of K such that K is algebraic over \sigma[K] is an automorphism of K, that is show \sigma[K]=K.

    I know \sigma^{-1}:\sigma[K] \rightarrow K can be extended to an isomorphism \mu:K\rightarrow K' where K'<=K. And since K<=K'<=K we know \sigma^{-1} can only be extended to an automorphism of K. But does this help me? I don't see how to make the connection with \sigma[K] .

    Any advice would be great! :-)
    Hint: show that \sigma(K) is algebraically closed too and thus, since K is algebraic over \sigma(K) \subseteq K, we must have \sigma(K)=K.
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