Describe, up to isomorphism, all the \(\mathbb{R}[x]\)-module structures one might put on a 3-dimensional real vector space (extending the fixed \(\mathbb{R}\)-action).
Solution
1. An \(\mathbb{R}[x]\)-module structure on \(V\) extending its given \(\mathbb{R}\)-vector-space structure is equivalent to a linear operator \(T\in\operatorname{End}_{\mathbb{R}}(V)\).
Given such a module structure, define \(T(v)=x\cdot v\). Since the \(\mathbb{R}\)-action is fixed, \(T\) is \(\mathbb{R}\)-linear, and then \[ p(x)\cdot v=p(T)v \] for every \(p\in\mathbb{R}[x]\). Conversely, any \(T\in\operatorname{End}_{\mathbb{R}}(V)\) defines an \(\mathbb{R}[x]\)-action by this formula. An isomorphism \(V_T\to V_{T'}\) is exactly an invertible \(\mathbb{R}\)-linear map \(P\) satisfying \(PT=T'P\), so isomorphism classes are similarity classes of operators on \(\mathbb{R}^3\).
2. As an \(\mathbb{R}[x]\)-module, \(V\) is finitely generated torsion of total elementary-divisor degree \(3\).
A real basis of \(V\) also generates \(V\) as an \(\mathbb{R}[x]\)-module, so \(V\) is finitely generated. Let \(\chi_T(x)\) be the characteristic polynomial of \(T\). By Cayley–Hamilton, \[ \chi_T(T)=0, \] so \(\chi_T(x)\) annihilates all of \(V\); hence \(V\) is torsion. Since \(\mathbb{R}[x]\) is a PID, the elementary-divisor theorem gives \[ V\cong\bigoplus_j \mathbb{R}[x]/(p_j^{e_j}), \] with \(p_j\) monic irreducible. As real vector spaces, \[ \dim_{\mathbb{R}}\mathbb{R}[x]/(p_j^{e_j})=e_j\deg p_j, \] so \[ \sum_j e_j\deg p_j=3. \]
3. The only monic irreducibles over \(\mathbb{R}\) have degree \(1\) or \(2\).
Every real polynomial of odd degree has a real root. Thus an irreducible real polynomial has degree at most \(2\). The monic irreducibles are the linear factors \(x-\lambda\) and the quadratics \[ x^2+ax+b, \qquad a^2-4b<0. \]
4. Therefore every \(3\)-dimensional \(\mathbb{R}[x]\)-module is isomorphic to exactly one module from the following four families, modulo the evident permutation of direct-sum factors.
4.1. Three linear elementary divisors:\[ \mathbb{R}[x]/(x-\lambda_1) \oplus \mathbb{R}[x]/(x-\lambda_2) \oplus \mathbb{R}[x]/(x-\lambda_3), \]with \(\lambda_i\in\mathbb{R}\).
4.2. One squared linear divisor and one linear divisor:\[ \mathbb{R}[x]/((x-\lambda)^2) \oplus \mathbb{R}[x]/(x-\mu), \]with \(\lambda,\mu\in\mathbb{R}\).
4.3. One cubed linear divisor:\[ \mathbb{R}[x]/((x-\lambda)^3), \]with \(\lambda\in\mathbb{R}\).
4.4. One irreducible quadratic divisor and one linear divisor:\[ \mathbb{R}[x]/(q(x)) \oplus \mathbb{R}[x]/(x-\lambda), \]where \(q(x)=x^2+ax+b\) satisfies \(a^2-4b<0\) and \(\lambda\in\mathbb{R}\).
4.4. There is no other possibility because irreducibles over \(\mathbb{R}\) have degree only \(1\) or \(2\).The elementary-divisor theorem also gives uniqueness of the multiset of elementary divisors, hence these families classify the module structures up to isomorphism.
5. In operator language, the four families are respectively: three \(1\times1\) real Jordan blocks; a \(2\times2\) real Jordan block plus a \(1\times1\) block; one \(3\times3\) real Jordan block; or one real \(2\times2\) block for a nonreal conjugate pair plus one real \(1\times1\) block.
These are precisely the rational/Jordan forms associated to the elementary divisors listed in 4.