Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 4 / 11G (Groups, Rings and Modules)

The structure theorem for finitely generated modules over a principal ideal domain says that a finitely generated module over a Euclidean domain $R$ iswith the rank and nonunit invariant factors of a linear operator unique up to associates.

Make $F^n$ an $F[X]$-module by $Xv=Av$. It is finitely generated and torsion, so the structure theorem gives cyclic summands $F[X]/(d_i)$ with $d_1\mid \cdots \mid d_k$. In the power basis of each summand, multiplication by $X$ has the companion matrix of $d_i$. Their block diagonal sum is the rational canonical form, unique because the invariant factors are unique.

The characteristic polynomial is $(t-\frac{1}{2}{)}^3$. For $N=A-\frac{1}{2}I$, one finds $N\ne 0$ but $N^2=0$, so the minimal polynomial is $(t-\frac{1}{2}{)}^2$. Hence the invariant factors are $t-\frac{1}{2}$ and $(t-\frac{1}{2}{)}^2$, and the rational canonical form is