Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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The vector annihilator is the unique monic generator of Ann⁡T(v) and divides the minimal polynomial

Statement

For an endomorphism T of a finite-dimensional vector space and v∈V, Ann⁡T(v) is a nonzero ideal of F[x]. It has a unique monic generator mT,v, and p(T)v=0⟺mT,v∣p. Moreover mT,v∣μT. In particular mT,0=1.

Facts & Assumptions

Given: A finite-dimensional endomorphism T and a vector v.

[L1]

The annihilator ideal is Ann⁡T(v)={p:p(T)v=0} (Cyclic subspaces, cyclic vectors, and vector annihilators).

[L2]

The minimal polynomial satisfies μT(T)=0 (The annihilator ideal is nonzero and has a unique monic generator; p(T)=0 if and only if μT∣p).

[L3]

Every ideal of F[x] is principal (For every field F, F[x] is a principal ideal domain).

Proof

technique · direct
1.1L1L2algebra

Linearity of polynomial evaluation shows that [L1] is closed under addition and under multiplication by arbitrary polynomials, so it is an ideal; it is nonzero because [L2] puts μT in it.

2.1step 1.1L3algebra

By [L3] the ideal is (a) for a nonzero polynomial a; scaling by the inverse of its leading coefficient gives a monic generator mT,v, and two monic generators of one ideal are associates and hence equal.

3.1step 1.1step 2.1L1L2∎

Membership in (mT,v) is exactly divisibility by mT,v, proving the displayed equivalence. Since μT belongs to the ideal, mT,v∣μT; if v=0, the ideal contains 1 and its monic generator is 1.

Depends on

Used by

Cited to discharge well-definedness by Cyclic subspaces, cyclic vectors, and vector annihilators.

Dependency tree · two levels

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Sources