DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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

Definition

Let $T : V \to V$ be an endomorphism of a finite-dimensional vector space over $F$ . Its annihilator set is

$Ann (T) : = {p \in F [x] : p (T) = 0},$

where polynomial evaluation at $T$ is that of Polynomial evaluation at an endomorphism: $p (T) = \sum_{k} a_{k} T^{k}$ .

Once The annihilator ideal is nonzero and has a unique monic generator; $p (T) = 0$ if and only if $μ_{T} ∣ p$ ↗ proves that this set is a nonzero principal ideal, the minimal polynomial of $T$ , denoted $μ_{T}$ , is its unique monic generator (The ideal generated by a subset and principal ideals, Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree). Thus existence is not built into the definition. On the zero space, $I_{V} = 0$ , so $1 \in Ann (T)$ and $μ_{T} = 1$ .

Depends on

Used by

The annihilator ideal is nonzero and has a unique monic generator; p(T)=0 if and only if μ_T∣ p Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 4 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Keith Conrad, The Minimal Polynomial and Some Applications, §4 (standard reference, not scraped)
Sheldon Axler, Linear Algebra Done Right, 4th ed., §5B (standard reference, not scraped)

The annihilator set Ann⁡(T)={p∈F[x]:p(T)=0}Ann(T)={p∈F[x]:p(T)=0}; once existence is proved, its unique monic generator μTμT​ is the minimal polynomial