isomorphism extension theorem

• April 13th 2009, 07:35 PM
ziggychick
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! :-)
• April 13th 2009, 09:48 PM
NonCommAlg
Quote:

Originally Posted by ziggychick
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 $\sigma(K)=K.$