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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 ]

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up vote 6 down vote accepted

In general, if $A \subset B$ is an extension of rings, I would call $B^{\times}/A^{\times}$ the relative unit group. (I am probably not the only one, but I couldn't say how widespread this is.) I feel reasonably confident that there is no specialized terminology in the case of nonmaximal orders.

To answer your non-terminological question: yes, if $A \subset B$ are orders in the same number field, the group $B^{\times}/A^{\times}$ is finite. This follows from the fact that one can prove the Dirichlet Unit Theorem equally well for a nonmaximal order $A$ in a number field $K$: $A^{\times}$ is still finitely generated with torsion subgroup equal to the roots of unity in $A$ and free rank equal to $r+s-1$, where $r$ is the number of real places and $s$ is the number of complex places of $K$. Thus we have finitely generated abelian groups $A^{\times} \subset B^{\times}$ with the same free rank, so $B^{\times}/A^{\times}$ is finite.

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Thanks! I tried "relative unit group" in my searches, but it kept coming up with the kernel of the norm, which I think is different. It also came up with weirder definitions, but never this one that I found. Do you know how I could calculate this relative unit group for an order A in the ring of integers B of a nice number field (of low degree)? I am curious if this "1 or 3" has any reasonable analogues for cubic or biquadratic fields, but I have no idea how to calculate a single example. – Jack Schmidt Feb 15 '11 at 3:59
And just to make sure I understood your "no specialized terminology": you do explicitly mean that [U(B):U(A)] is not some standard invariant of an order (like the class number or something equally famous). It seems natural in the quadratic case, but I guess in general there is no "one true" minimal polynomial for a field. – Jack Schmidt Feb 15 '11 at 4:08
@Jack: about "relative unit group": I checked MathSciNet, and there were only 3 hits, all of which seemed to refer to something else (still involving unit groups of extensions of number fields). No, I don't know any especially good way to calculate this group. My own personal experience with nonmaximal orders is limited to the case of quadratic fields, where things are much simpler. Finally, yes, I explicitly mean that the order of the quotient is not some standard, well-studied invariant of a nonmaximal order, as far as I know. – Pete L. Clark Feb 15 '11 at 16:37
Sounds good to me. The unit theorem at least tells me an answer exists, and the rest of your answer tells me I'm not simply missing the name of some well-known Kash or Magma command that calculates this using class field theory or something. – Jack Schmidt Feb 15 '11 at 16:52

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