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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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. 


If K and L are Ffield extensions, K/F and L/F are both finite dimensional, and the isomorphism from K to L is an Fhomomorphism, then the proof is easy, but the general case seems difficult. 

