Conjugacy class in $\mathrm{GL}(n,\mathbb{Z})$ of a matrix related to cyclotomic polynomials

Question

I have a finitely generated subgroup $X$ of $(\mathbb{C},+)$ that is invariant under multiplication by a primitive $d$th root of unity $\zeta_d$, for some $d\geq 1$.

Since $\mathbb{C}$ is abelian and torsion-free, $X\cong \mathbb{Z}^n$ for some $n$ and $\zeta_d$ can be viewed as an element of $\mathrm{GL}(n,\mathbb{Z})$.

Question

• Can we choose a basis of $\mathbb{Z}^n$ so that the matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ corresponding to $\zeta_d$ is a block matrix of the form $$A=\begin{bmatrix} \bf{M_d} & * \\ \bf{0} & * \end{bmatrix}$$ where $\bf{M_d}$ is the companion matrix of the cyclotomic polynomial $\Phi_d$?

Equivalently, one can first fix any basis of $\mathbb{Z}^n$, express $A$ with respect to this basis, and then ask if $A$ is conjugate in $\mathrm{GL}(n,\mathbb{Z})$ to a block matrix as above.

My work so far

Since all orbits of $\zeta_d$ on non-zero elements of $X$ have length exactly $d$, the characteristic polynomial of $A$ is $(\Phi_d)^k$ for some $k\geq 1$. (In particular, $n=\phi(d)k$.) Since $\Phi_d$ is irreducible over $\mathbb{Z}$, $A$ is conjugate in $\mathrm{GL}(n,\mathbb{Z})$ to a block-triangular matrix, with all blocks on the diagonal having characteristic polynomial $\Phi_d$. (I found this result in "Integral Matrices" by Newman.) So it remains to decide whether the top left block is conjugate to $\bf{M_d}$.

While researching this question, I learned about the Latimer-MacDuffee Theorem which, if I understand correctly, tells me that the number of conjugacy classes of matrices with characteristic polynomial $\Phi(d)$ is equal to number of ideal classes in the cyclotomic integers $\mathbb{Z}[\zeta_d]$ of the cyclotomic field $\mathbb{Q}(\zeta_d)$. Here I must admit that I am a bit out of my depth and don't know much about ideal classes and so on. I did learn that these class numbers can eventually become very large, so I cannot conclude that $A$ is conjugate to the desired form from the characteristic polynomial alone.

In Keith Conrad's very nice notes "Ideal classes and matrix conjugation over $\mathbb{Z}$" (https://kconrad.math.uconn.edu/blurbs/gradnumthy/matrixconj.pdf), Corollary 2.8 gives a criterion to determine whether a matrix with an irreducible characteristic polynomial is conjugate in $\mathrm{GL}(n,\mathbb{Z})$ to the companion matrix of its characteristic polynomial. I'm having a bit of trouble deciding whether this result can help as I'm not familiar with the terminology, especially fractional ideals. I'm still working on trying to unpack all of this, but I'd welcome help.

I guess the question is whether there is extra information in the original setup (the embedding of $X$ in $\mathbb{C}$, etc.) that can be leveraged to apply Conrad's Corollary 2.8? Or perhaps there is an alternate more direct proof? Or perhaps the answer to my question is no, and in fact nothing more can be said about $A$ apart from its characteristic polynomial?

Answer 1

No (edit: for $k = 1$, but yes for $k \ge 2$, see the comments). It's not necessary to appeal to the Latimer-MacDuffee theorem in this special case and we can just use standard results in algebraic number theory. $X$ is a torsion-free module over $\mathbb{Z}[\zeta_d]$, and in particular can be a module corresponding to any nontrivial ideal class in $\mathbb{Z}[\zeta_d]$ (since $\mathbb{Q}(\zeta_d)$ embeds into $\mathbb{C}$); these exist whenever the class number is greater than $1$, which first occurs at $d = 23$. Explicitly, we can take

$$I = \left( 2, \frac{1 + \sqrt{-23}}{2} \right) \subset \mathbb{Z}[\zeta_{23}].$$

$I$ is a module of rank $1$ over $\mathbb{Z}[\zeta_{23}]$, so it is isomorphic to a free $\mathbb{Z}[\zeta_{23}]$-module iff it is principal, which is known to not be true (in general the ideal class group of a number field $K$ can be identified with the group of isomorphism classes of rank $1$ torsion-free modules with respect to tensor product). See also the structure theorem for f.g. modules over a Dedekind domain for more on this (this theorem will classify the possible actions of $\zeta_d$ on $X$).

Working through the definitions this is equivalent to saying that the action of $\zeta_{23}$ on $I$ defines a matrix which is not conjugate to the companion matrix of $\Phi_{23}(x)$.