For $A=\mathbb{Z}[x]/(f)$ with quotient field $K$ and ring of integers $B$, does $U(B)/U(A)$ have a name?
For instance $u = \tfrac{1+\sqrt{5}}{2}$ is a unit in $\mathbb{Q}[\sqrt{5}]$, but neither $u$ nor $u^2$ has integer coefficients in the basis $\{ 1, \sqrt{5} \}$. Of course $u^3$ has integer coefficients (spooky if you haven't tried it!) and in fact $u^n$ has integer coefficients iff $0 \equiv n \mod 3$.
For quadratic fields with basis $\{ 1, \sqrt{n} \}$ for $n$ square-free, one almost always has $U(A) = U(B)$. If not, then $[ U(B) : U(A) ] = 3$.
That's crazy, and it should have a name. For instance, I'd like to find out if the following is true, but I don't even know what to look for:
Is $U(B)/U(A)$ always finite? [ where $B$ is the ring of integers of an order $A$ in a number ring ]