Let K and L are fields,L is a sub field of K,and L is isomorphic to K,whether can we get K=L?If true,how to prove? Thanks.
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closed as too localized by Andreas Blass, Franz Lemmermeyer, Emil Jeřábek, Leonid Positselski, Andres Caicedo Nov 26 at 6:53 |
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No. ${\mathbb C}(X^2,Y)=L$ is a subfield of $K={\mathbb C}(X,Y)$ where $X,Y$ are algebraically independent variables over $\mathbb C$. Hence $L$ is isomorphic to $K$ but not equal. |
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If K and L are F-field extensions, K/F and L/F are both finite dimensional, and the isomorphism from K to L is an F-homomorphism, then the proof is easy, but the general case seems difficult. |
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