I have been struggling with the following problem: 

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> How can one prove that if there is an isomorphism between several variable polynomial rings over a field $K$, $ \varphi : K[X\_1, \dots, X\_n] \to K[X\_1, \dots, X\_m]$ such that $\varphi$ restricted to $K$ is $id$, then one must have $m=n$. 
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Any help appreciated, thanks.