This is old but never really received satisfactory treatment.
lime wrote:x -> 2x is also "onto" and "one-to-one", isn't it?
Yes, but it doesn't preserve the multiplicative structure of
f(x+y) = f(x) + f(y) for x,y ε Q iff for ax + b, b = 0; also, if f is just an additive group isomorphism from Q onto Q, then a would trivially be 1. Isn't this a practice test question? I recall originally thinking Q had ∞ automorphisms then rethinking that it must only be 1.
All automorphisms of
are linear transformations of the form
However, since we're talking about ring automorphisms, we have
and hence the result. More interestingly,