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The annihilator ideal is nonzero and has a unique monic generator; $p (T) = 0$ if and only if $μ_{T} ∣ p$

Statement

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 $μ_{T}$ . For every $p \in F [x]$ ,

$p (T) = 0 ⟺ μ_{T} ∣ p .$

For the zero space, $μ_{T} = 1$ .

Facts & Assumptions

Given: An endomorphism $T : V \to V$ of a finite-dimensional vector space over a field $F$ , and the annihilator set 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 field $F$ , every ideal of $F [x]$ is principal (For every field $F$ , $F [x]$ is a principal ideal domain).

[L2]

Cayley–Hamilton states $χ_{T} (T) = 0$ for every finite-dimensional endomorphism $T$ (Cayley-Hamilton: every finite-dimensional endomorphism satisfies its characteristic polynomial, $χ_{T} (T) = 0$ ).

[L3]

For $A \in M_{n} (F)$ the polynomial $χ_{A}$ is monic of degree $n$ , and $χ_{A} = 1$ when $n = 0$ ( $χ_{A} (x)$ is monic of degree $n$ ; for $n \geq 1$ its $x^{n - 1}$ coefficient is $- tr (A)$ and its constant coefficient is $(- 1)^{n} \det (A)$ , while $χ_{0 \times 0} = 1$ ); and $χ_{T}$ is by definition $χ_{[T]_{B}^{B}}$ for any ordered basis $B$ , independently of the choice (The basis-independent characteristic polynomial $χ_{T}$ of an endomorphism of a finite-dimensional space, including $χ_{T} = 1$ in dimension zero).

[L4]

Every nonzero polynomial has a leading coefficient and is monic exactly when that coefficient is $1$ (Degree, leading coefficient and monic polynomial, with the zero polynomial having no degree).

Proof

technique · direct

1.1

The zero polynomial lies in $Ann (T)$ . If $p (T) = q (T) = 0$ , then $(p - q) (T) = 0$ ; and if $h \in F [x]$ and $p (T) = 0$ , distributivity of the finite polynomial sums and associativity of composition give $(h p) (T) = h (T) p (T) = 0$ . Hence $Ann (T)$ is an ideal.

givenalgebra

1.2

By [L2], $χ_{T} \in Ann (T)$ . Fix an ordered basis $B$ of the $n$ -dimensional $V$ ; by [L3], $χ_{T} = χ_{[T]_{B}^{B}}$ with $[T]_{B}^{B} \in M_{n} (F)$ , so $χ_{T}$ is monic of degree $n$ and in particular nonzero. Thus the ideal is nonzero, including when $V = 0$ , where $χ_{T} = 1$ .

L2L3

1.3

If $u$ and $v$ are monic generators of this nonzero ideal, then $u = a v$ and $v = b u$ for some polynomials $a, b$ . Degrees force $a, b$ to be nonzero constants, and monicity forces both constants to be $1$ ; hence $u = v$ .

L4algebra

2.1

By [L1], write $Ann (T) = (g)$ with $g \neq 0$ . Multiplying $g$ by the inverse of its leading coefficient gives a monic generator $μ_{T}$ by [L4], and step 1.3 shows that this monic generator is unique.

step 1.2step 1.3L1L4choose

3.1

Finally, $p (T) = 0$ means $p \in Ann (T) = (μ_{T})$ , which is equivalent to $p = μ_{T} q$ for some $q \in F [x]$ , that is, $μ_{T} ∣ p$ . When $V = 0$ , step 1.2 gives $(1) = F [x]$ , so its monic generator is $1$ .

step 1.2step 2.1given∎

Depends on

Used by

Cited to discharge well-definedness by The annihilator set Ann(T)={p∈ F[x]:p(T)=0}; once existence is proved, its unique monic generator μ_T is the minimal polynomial.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 63 results over 10 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, Theorem 4.4 (standard reference, not scraped)
Sheldon Axler, Linear Algebra Done Right, 4th ed., §5B (standard reference, not scraped)

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