let k be a field.

a) show that the mapping $\displaystyle \varphi : k[t] \to k[t]$ definded by $\displaystyle \varphi(f(t))=f(at+b)$ for fixed $\displaystyle a,b \in k,a \ne 0$ is an automorphism of k[t] which is the identity on k.

b) conversely, let $\displaystyle \varphi$ be an automorphism of k[t] which is the identity on k. Prove that thre exist $\displaystyle a,b \in k $ with $\displaystyle a \ne 0$ such that $\displaystyle \varphi(f(t)=f(at+b)$ as in part a)

So I have done part a) but I can't seem to get out of the gate for part b).

I keep reading the problem and only seeing the trivial solution.

The identitiy map is an automorphism so $\displaystyle a=1$ and $\displaystyle b=0$ satisfy the above, but I have a feeling that is not what Dummit and Foote had in mind.

A push in the right direction would be great.

Thanks

TES