Let $A$ be a domain and $B = A[T, \frac1T]$ for an indeterminate $T$. Prove that the ring automorphism group $Aut(B_{/A})$ of $B$ inducing the identity on $A$ is finite if and only if the group of invertible elements of $A$ is finite.
I want to say that the automorphisms of $B$ inducing the identity on $A$ is $Gal(B/A)$. It remains to show that $|Gal(B/A)| < \infty$ iff the group of invertible elements of $A$ is finite. The order of the Galois group is equal to the degree of a normal extension (or divides any extension). However, I'm not sure how to find (or just upper bound) $|Gal(B/A)|$.