Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 11G (Groups, Rings and Modules)

A Smith normal form of an $m\times n$ matrix $A$ over a principal ideal domain $R$ is a diagonal matrixwhere $U\in G{L}_m(R)$ and $V\in G{L}_n(R)$. It is guaranteed to exist for every matrix over a principal ideal domain. The entries $d_i$ are determined up to multiplication by units; over $\mathbb{Z}$ they are conventionally chosen positive.

Present a finitely generated abelian group $G$ as the cokernel of an integer matrix $A:{\mathbb{Z}}^m\to {\mathbb{Z}}^n$. Changing $A$ by the invertible row and column operations used in its Smith normal form does not change the isomorphism type of the cokernel. Ifthenwhere factors with $d_i=1$ may be omitted. This is the Fundamental theorem of finitely generated abelian groups; the rank of an abelian group $r$ and invariant factors $d_i>1$ are unique.

There aresuch conjugacy classes. Every matrix in $G{L}_{10}(\mathbb{Q})$ is an invertible matrix, so by the minimal polynomial of an invertible matrix its minimal polynomial must have nonzero constant term. Buthas constant term zero and cannot be the minimal polynomial of an invertible matrix.