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Thread: 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 $\displaystyle \sigma$ of K onto a subfield of K such that K is algebraic over $\displaystyle \sigma[K] $ is an automorphism of K, that is show $\displaystyle \sigma[K]=K$.

    I know $\displaystyle \sigma^{-1}:\sigma[K] \rightarrow K $ can be extended to an isomorphism $\displaystyle \mu:K\rightarrow K'$ where K'<=K. And since K<=K'<=K we know $\displaystyle \sigma^{-1}$ can only be extended to an automorphism of K. But does this help me? I don't see how to make the connection with $\displaystyle \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 $\displaystyle \sigma$ of K onto a subfield of K such that K is algebraic over $\displaystyle \sigma[K] $ is an automorphism of K, that is show $\displaystyle \sigma[K]=K$.

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

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