The only field automorphism is the identity. This can be seen since or , but the latter does not give an injective map, since ; then, for integers ; and thus, for integers , , since , we have .
Is there a solution manual that says there are two field automorphisms on the rationals? That's wrong.
Expanding on this, nontrivial field automorphisms have to fix some proper subfield, but the rationals don't contain subfields -- its own prime subfield (which is essentially what cincodemayo5590 shows above).