PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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For finite-dimensional $V$ , $V_{T}$ is finitely generated and torsion, with annihilator generated by the minimal polynomial

Statement

Let $T$ be an endomorphism of a finite-dimensional $F$ -vector space $V$ . The $F [x]$ -module $V_{T}$ is finitely generated and torsion. The annihilator of $V_{T}$ is generated by the minimal polynomial $μ_{T}$ .

Facts & Assumptions

Given: The polynomial action of The $F [x]$ -module $V_{T}$ of an endomorphism, the module finite-generation convention of Generated submodule, cyclic and finitely generated modules, module basis and free module, and the operator-annihilator definition of The annihilator set $Ann (T) = {p \in F [x] : p (T) = 0}$ ; once existence is proved, its unique monic generator $μ_{T}$ is the minimal polynomial.

[L1]

For every endomorphism $T$ of a finite-dimensional $F$ -vector space, $Ann (T)$ is a nonzero ideal of $F [x]$ and has a unique monic generator $μ_{T}$ ; moreover $p (T) = 0$ exactly when $μ_{T} ∣ p$ (The annihilator ideal is nonzero and has a unique monic generator; $p (T) = 0$ if and only if $μ_{T} ∣ p$ ).

Proof

technique · direct

1.1

Any finite $F$ -basis of $V$ generates $V_{T}$ over $F [x]$ , because constant polynomials already give every $F$ -linear combination. On the zero space the empty basis generates.

given

1.2

By [L1], $μ_{T} (T) = 0$ , so $μ_{T}$ kills every vector. Thus every vector is torsion and $V_{T}$ is a torsion $F [x]$ -module.

2.1

A polynomial $p$ annihilates the whole module exactly when $p (T) v = 0$ for every $v$ , exactly when $p (T) = 0$ . By [L1], this is equivalent to $p \in (μ_{T})$ , so ${Ann}_{F [x]} (V_{T}) = (μ_{T})$ . For $V = 0$ , both ideals are $(1)$ .

L1given∎

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Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Brussel, Finitely Generated Modules over a PID, Section 5.2 (standard reference, not scraped)
A. Apisa, Wisconsin Math 542, Lectures 11-12 (standard reference, not scraped)

For finite-dimensional VV, VTVT​ is finitely generated and torsion, with annihilator generated by the minimal polynomial