Is it true that a field automorphism must fix the elements of any proper subfield? Ie, since $\mathbb{Q}\subset\mathbb{R}$, all automorphisms from R to itself must preserve the map from $I:\mathbb{Q}\to\mathbb{Q}$?
note: I know that there is only one automorphism of $\mathbb{R}$ --- I am more curious to see if the reasoning above is correct.